Since last year, I've been interested in higher mathematics and don't really know where to start and don't know what knowledge I still need to obtain, before being able to understand the concepts. Currently, I'm in a trigonometry class (I understand it well), but I have studied other subjects by reading textbooks and taking free courses, when possible. If I were to sum up my knowledge, up until trigonometry, I have a good understanding of the topics. I'm also good with symbollic logic. After that point, I have a fragmented knowledge of the topic. I want to study logic, set theoy, abstract algebra, etc. and topology (the most seemingly interesting subject to me). Where do I start? What subject should I start with? What lower mathematics should I learn first?
There can't be more said to make sure you have the fundamentals down very well. Do your best to understand the topics well.
Having said that, if you're ready for more, there's plenty of places to go. It depends on your interest and your background.
One way to go would be to try to accelerate yourself through the standard Trigonometry, Geometry, Pre-Calculus, Calculus curriculum. This is a good way to go, but won't be the most direct way to the more abstract topics you've listed. If this is of interest, perhaps "A Hitchhiker's Guide to Calculus" by Spivak would be a good place to start.
A good place to start might be number theory. You already have some feelings for numbers, and it'll be a great way to introduce yourself to proving things rigorously. For number theory, this link might be of interest: Best book ever on Number Theory.
Alternatively, you can never start learning to prove things rigorously (essentially thinking clearly) too early. The text "Foundations of Higher Mathematics" by Fletcher and Patty might be a good place to start.
Best of luck!
I think people should have a background in analysis and algebra before attempting to learn serious set theory and logic. Those subjects tend to be a bit too abstract to be the place where you first learn to reason mathematically. (I'm not talking about the rudiments of set theory that are a prerequisite for analysis and algebra.)
I think in your shoes I would start learning calculus with Spivak's book Calculus. That is an excellent introduction to mathematics in general, not just calculus. It has a solutions manual, which you should use mainly to check your own solutions.
If you find that book too difficult at first, you could try reading the high-school level books in the Gelfand series (The Method of Coordinates, Functions and Graphs, Algebra and Trigonometry). They have problems that are probably more challenging than what you're doing in school. You could also have a look at some of the books in the New Mathematical Library. They are written on various topics and directed at bright high-schoolers. The books by Niven in particular are good. After a while learning from these kinds of books, Spivak's ought to seem easier.
If you do want to jump straight into abstract algebra, that might be difficult, but you could try Algebra by Artin. You could also read Stark's Introduction to Number Theory. Learning topology in a meaningful way is probably impossible at this stage.
An alternative to all of this, if you want to get an overview of various mathematical topics, is to read What is Mathematics? by Courant and Robbins.