Linear algebra normally concerns linear systems, which would normally involve Matrices (Matrix math), vectors, vector spaces, etc. It is a bit involved since normally a good deal of the material relies on proofs and not simply memorization and computations.
Abstract Algebra is a more modern idea I believe and relies on the actual study of algebraic concepts, this is normally beyond even Linear Algebra, but is not really "advanced" material in the normal way. It requires a good amount of *gasp* abstract concepts (hence the name), since it's attempting to define an entire mathematical model and how/why it works and can be extended to other ideas. So again, this uses a lot of proofs and abstract ideas.
Algebra I and II can have their fair share of proofs, but in general, they rely on more basic algebraic concepts, their applications, etc.
Linear Algebra is normally taken, or able to be taken, alongside Calculus I; though it is suggested that the student have a decent grasp of proofs and/or formal logic (normally taught in discrete math courses).
Abstract Algebra would likely fit into that category as well, and an undergraduate/introductory course would probably not be too difficult for a student that can tackle Calculus/Linear Algebra.
As far as the College Algebra, yes, it would be suggested that you know College Algebra before taking Linear and Abstract Algebra, but Linear Algebra is kind of in a league of it's own as well as Abstract, they're just... different. A lot of their material depends on algebraic concepts, but it's not as if you need to be a "professional" mathematician to start learning them.
In both cases though there normally isn't too much Trigonometry or Geometry, BUT that doesn't mean there isn't the occasional problem/idea involving it or linking it. I would just suggest you go peruse a bit of the material to get your feet wet, there's quite a few free pdf's on the web (as well as Khanacademy.org of course). This is one I found with a quick google search that looks fine: https://www.math.ucdavis.edu/~linear/linear-guest.pdf
Again though, Calculus has a good deal of proofs and ideas, but linear and abstract algebra normally have a bit more, they're decently abstract in their ideas, and their underlying concepts are not what most people are used to (for example, matrix math is not commutative, I.E. given two matrices A and B then A * B IS NOT always equal to B * A, meaning order does matter.)