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I'm currently taking an introductory physics class, I hope to in the future take a more rigorous mechanics class. From what I've come across online, it seems that a lot of mechanics uses linear algebra, vector calculus, and differential equations. Is learning these three all that I'd need to make basic mechanics accessible?

For example, I want to be able to solve things such as the orbit equation (https://physics.stackexchange.com/questions/56657/how-is-the-equation-of-motion-on-an-ellipse-derived), or derive the equation for a multidimensional oscillator myself, or discuss electric fields in terms of vectors.

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    You can have a look at the book "Mathematical Methods of Classical Mechanics" by Arnold to get an idea what concepts are used. I'd say differential geometry (tangential spaces, differential forms, ...) and differential equations (ODEs, Hamiltonian systems, calculus of variations, ...) are a must. – 2017-01-20
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    You may want to also look up Lagrangian Mechanics. – 2017-01-20
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    For coupled oscillators, you need linear algebra also. – 2017-01-20

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For any advanced classical mechanics (Lagrangian/Hamiltonian mechanics) you need to learn calculus of variations. What you've listed is generally what is done in an undergraduate mechanics course which usually only requires some simple vector algebra and ordinary differential equations. The results seen in a rigorous course in Newtonian mechanics are usually derived from solving Newton's 2nd law (a 2nd order ODE) under different conditions, different coordinate systems, etc. If what you've listed is the kind of thing that you're interested in, I would focus on studying ordinary differential equations as from my experience the linear algebra used is quite minimal. If you're wanting to learn classical electrodynamics, that's a different story and vector calculus is essential. Undergraduate classical mechanics is not formulated as a field theory where as electrodynamics is. Hope this helps!

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Given your background information, for vector calculus I'd recommend:

David Bressoud, Second Year Calculus: From Celestial Mechanics to Special Relativity

H. M. Schey, Div, Grad, Curl, and All That: An Informal Text on Vector Calculus

Harry F. Davis, Introduction to Vector Analysis (but see my comments about it here)

As for topics in differential equations and linear algebra that you would want to learn, these are 2nd-3rd year courses (at U.S. universities) whose treatment is fairly standard, and thus most any of the widely used texts for them should be fine. Probably linear algebra is less important than differential equations (at least for the standard 2-semester upper level sequence in mechanics), and a combined course in the two subjects (if offered) would likely be enough as far as linear algebra is concerned.

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In order to introduce the Lagrangian and Hamiltonian, you need a little bit of calculus of variations, but most physics books explain the needed functional derivatives in anology to derivatives of functions, which suffices most physics students.

Afterwards, you will mostly solve the Newton, Hamilton or Euler-Lagrange equations (which are basically the same). Therefore you need to know how to solve ODEs.

Linear Algebra will be used to choose coordinate frames in which the physical problem simplifies or in order to solve ODEs.

If you are able to do calculations like surface and volume integrals, matrix diagonalisation to solve simple ODEs, I would recommend to simply start with any theoretical classical mechanics book and see if you have enough background.

Of course, there are different levels of understanding. For example, Hamilton formalism can be expressed in terms of symplectic manifolds (which needs much more mathematical background).