This section is meant to be an overview of subjects learned in a typical astrophysics degree program at the undergraduate level. Online resources and/or book recommendations will be made, use what you feel works best for you. How you obtain any books you may want to read is up to you. The exact materials, and depth of study for each area, will vary depending on the school attended. In general there are things every astrophysics student would be expected to have had some contact with though, and I will try to cover all of those. The areas of study will be broken up into four sections: Mathematics, Physics, Astronomy, and Miscellaneous. I would suggest having a firm understanding of mathematics up to at least basic calculus before attempting to go too far into the physics and astronomy work. These subjects rely heavily on mathematics, and a lack of understanding of their mathematical foundation will only make things harder for yourself. The miscellaneous section is for things that don't necessarily fall into the other three categories, but would be useful too know anyway. It's mostly optional but definitely worth going through if you can.
There are a few resources that are useful for any of the sections below, those are Khan Academy and MIT OpenCourseWare. Both of these are completely free and have a wealth of information covering many of the topics mentioned here. Khan Academy's strengths are in the video lectures it offers, while the MIT site's strengths are in the problem sets and lecture notes from real professors. Additionally, Open Culture offers an aggregate of online courses and textbooks from all over the web. This makes it much easier to browse for content that fits your needs. Another general use resource I would highly recommend is Wolfram Alpha. That site not only has informational pages, but has tools for handling computations and graphing. Not all features are free, but what it offered for free is very useful.
Each section is numbered (roughly) in the order they should be studied. As for how the entire guide should be followed, I would suggest something like this: Get comfortable with maths up to and including single-variable calculus. After that you can start on the physics, up through special relativity. You will need to learn multi-variable calculus at some point in there, so take care of that when you start seeing it. After special relativity you would be well prepared to start the astronomy sections. Some of the more advanced stellar astrophysics material will require quantum mechanics and thermodynamics, so it would be good to jump over to physics and learn those areas if you haven't already at that point, or save that material for later and go into galaxies and cosmology. Before quantum an thermodynamics you will want to learn linear algebra, statistics/probablility, and differential equations. Finish astronomy up to the general relativity section and that will cover most of what you need. That isn't a perfect outline, but it's a fairly reasonable one. Particle physics is great to know at least at a basic level, but isn't exactly essential for any of the other material. The same goes for chemistry. If you really want to test yourself after covering everything, the Physics GRE will tell you just how much you know, or don't know!
If this seems like a lot of work, that's because it is. There is no easy way to get a good understanding of this enormous field. It requires dedication and a lot of time. I suggest that if whatever resources you are using have practice problems, attempt as many as you can without looking up solutions first. Even if you get stuck, or get the wrong answer, the problem solving skills you develop from them over time will be invaluable. The ability to handle complex problems will not only aid you in your studies, by will also likely help you in your everyday life. Getting discouraged is natural, for every mountain you climb there will be an even larger one waiting ahead of you. I personally think gaining some true understanding of how our universe works, even if only a small amount, is worth it.
The basics of working with numbers. Review if necessary.
Working with lines and shapes in multiple dimensions. Review if necessary.
Methods for solving equations, graphing, abstraction of mathematical ideas, and much more. Essential before attempting any physics work.
The study of the relationships of side lengths and angles of triangles.
The study of change, both very small and large. Also offers a more general analysis of functions. Some familiarity with differential and integral calculus is recommended before starting physics work.
All about vector spaces. Recommended before doing any quantum mechanics work.
Dealing with uncertainty, data sets, and counting/selection methods. Probablility is recommended before doing any quantum mechanics work, and statistics before thermodynamics work.
Here are some things that will be helpful as you get deeper into your studies.
Differential equations: Methods for dealing with certain types of equations that show up all over the physical sciences. Calculus required.
Complex numbers: Number system beyond the real numbers. This has applications in various fields of physics so you will see it eventually.
Fourier analysis: Methods for analyzing and decomposiong waves.
Sequences and series: This will likely be covered in calculus, but if you miss it somehow it's worth checking out. See the Calculus section for links.
The physics of everything larger than atoms, essentially. A lot of fundamental concepts are learned here, such as position, velocity, acceleration, force, energy, and momentum. The main topics include vectors, Newtonian physics and equations of motion, rotational motion, vibrational motion, work and energy, Hamiltonians and Lagrangians. Another very important skill to learn here is dimensional analysis, which will be useful for the study of any scientific field. There is a lot here, and it's not mandatory to cover it all before moving on.
The physics of electricity and magnetism. You'll start off with the static (non-moving) cases of each force, then move into the more realistic dynamic study where the forces become interconnected. A key point is building up to a good understanding of Maxwell's equations.
The study of light and how it interacts with matter, with a focus on instrumentation for manipulating it. Typically, simplified models like geometric optics are used. It is recommended that you study this before looking into telescope designs and instrumentation.
The study of how space and time are interconnected.
The behavior of particles.
Thermodynamics is mainly the study of heat, and statistical mechanics is the theory used to describe the behavior of large and complex systems with high uncertainty (like a gas).
The field focused on all the particles that make up the Universe, their properties, and how to categorize them.
The "basics" covers things like astronomical coordinate systems, celestial mechanics, color and magnitude scales, blackbody radiation, spectral lines, and telescope designs. Essentially everything that is covered in the first six chapters of An Introduction to Modern Astrophysics by Carroll and Ostlie. That book is fantastic, and if you are very serious about learning astrophysics I would recommend getting a physical copy. Its main flaw is a lack of any solutions for the problems it has, unless you want to spend extra for the solutions manual. Another nice general use resource is Astronomy Notes which covers a lot of topics, although not always in much depth.
Everything about stars. Basic stellar properites. HR diagrams. Stellar classifications. Stellar formation and evolution. Stellar atmospheres and interiors. Energy production and transfer methods.
Galaxy types and characteristics. The structure of galaxies. Galaxy formation and evolution. Dark matter.
Cosmology studies the Universe as a whole. Its history and how it has evolved over time. Topics include the thermal history of the Universe, Inflation and the Big Bang, Hubble's law, the Friedmann equations, the formation of large scale structure, the Cosmic Microwave Background (CMB), and dark energy.
Even if you don't plan to ever do any observing yourself, it would be good to learn some of the basics about the technology used in astronomy. This includes telescope designs and the common instruments that are used to gather data. Methods used for observing would also be worth a look. There are ways to contribute to the field even as an ametuer astronomer if you care to invest in the right equipment, so if that is something that interests you this subject will be very important. One of the most important intruments to know about is the charged-coupled device (CCD), which is used for collecting light from astronomical sources.
Another important topic to be aware of is data reduction. When you gather data you don't just start doing your science on it right away, it fist has to be "cleaned up" and put into a more managable form. This is especially true for astronomy data, due to how easy it is for extra noise to get into your data. Astronomers take several different types of images along with their actual science targets to aid in the image reduction process. These extra images are typically "darks", "flatfields", and "bias" images which will be explained in the links below. The typical software used to reduce images is IRAF, and there is also a Python package that can be used along with that called PyRAF. This is a fairly advanced topic and won't be all that useful unless you plan to do some professional observing, but a basic overview of the topic should be done so you can appreciate all the work that goes into getting astronomy data.
This is an advanced topic and it is unlikely you would be expected to have more than a very basic idea of this subject at the level this guide is meant for. I only mention it because it is very important for several areas of astrophysics. If you have covered everything else and want a heavy challenge, this would be something to consider looking up.
A bit of general chemistry would be helpful to know before starting on stellar astrophysics and/or thermodynamics. Particularly the structure of atoms and molecules, moles, and the laws that govern chemical reactions.
If you were going to do any physics based research at a university, it is almost guaranteed you would need to know computer programming at some level. Even for independent learning, it would be a good idea to get some experience with it for doing calculations and making graphs. There are a lot of programming languages out there, but the first one you learn isn't that important. Once you know one, learning another isn't too difficult. Python is widely used in astronomy and fairly easy to work with so I would highly recommend that. Once you get some familiarity with it you can check the main page for links to additional python resources. IDL has been used a lot in astronomy in the past too, but its popularity has dropped and it requires a license to use, making it less appealing to learn these days. If you were wanting something more challenging but also more powerful, C or C++ are good choices, although graphing is better done with other languages or tools. Programming is also fun for non-physics based uses, like building your own website or making personalized applications.
Alternatively, there are software packages that you can get (at a price) for doing a lot of the work for you. These are Mathematica and Maple. Both are very powerful and useful, although due to their proprietary nature they have some limitations that writing your own programs do not. There is also Matlab, which is widely used in mathematics and physical sciences.
A set of instructions that clearly define how to do something. This is mainly additional information for the computer programming section. If you get into programming you will likely be writing your own algorithms eventually, so some formal study of them could be useful. A more in depth study of computer science as a whole would be ideal if you were wanting to get heavily into this area.
Often, mathematics (and even physics/astronomy) textbooks will ask you to prove something is true. At the level of work for this guide this typically will just require you to use a few definitions and rearrange one side of an equation to be equal to the other side, thus proving the desired statement is true (or proving the statement is not true, depending on the question). Things are much more complicated than that however. If you are interested in getting to know mathematical logic and how we know what is true in mathematics on a deeper level, you may want to do some additional reading into the subject. It could also help when reading textbooks, as proofs are used all the time in them, and knowing how proofs work could make studying many topics easier to understand.
If you were wanting to go to graduate school in America for physics/astronomy, you would most likely be required to take the Graduate Record Examination (GRE), specifically the general test and the subject test for physics. Your score on these would be one of the factors schools judged you on when deciding if they would admit you into their graduate program or not. The actual efficiency of this test for deciding how a student will do in their graduate education is up for debate. One thing is certain however, the physics GRE is hard. Really hard. I wouldn't recommend anyone willingly take this thing, but if you are wanting the ultimate test of how much physics you can recall and use correctly in a single sitting, this is the test to take. There are several of the older exams from past years freely available online if you are interested in taking it on. A note about the difficuly though; some questions are purely "you know it or you don't" trivia type questions. Things like that don't test your ability to actually do physics, so it is important to pay attention to the types of questions you are getting correct or incorrect, and not just your raw score.