The best way to improve your maths skills are;
1. practice
2. develop a deeper understanding
Both can be obtained through using textbooks correctly (that means taking notes through the book and doing all the problems)
In terms of what you should know before you go to college/uni.. I wouldn't say there is a limit to how much you should know, the more you know the easier your math classes will be (which will be a blessing since I've heard a LOT of stories about maths being very poorly taught) and if you keep up your rate of learning you'll constantly be ahead!
I use to think that I wasn't good at learning from textbooks either. It turned out I just wasn't using textbooks correctly and that I was hand waving past the problems. Learning from textbooks is an acquired skill, yes, but it is perfectly achievable and it's a very useful skill to have. It was either that or I had some very poorly written textbooks.
Take your textbook, read a section, take notes on that section (irl notes on a piece of paper with a pen), read the notes at the end of each section and make sure you understand them. Continue onto the next section and repeat until you land at the problems. Attempt every problem, spend as much time as you need to completely every last one before moving onto the next chapter. Repeat until you have finished the book.
Another set of things you should learn are derivations, learn how things are derived, don't just go straight for remembering formulas for things, learn how they got there, learn to do them without having to look at notes, you can think of these derivations as the forms you learn if you've ever done a martial art :p
I'll give you the list of books and material I've used in my first year of learning maths (after you've gone through everything you should know exactly where to go to continue your journey)
1)Introduction to Linear Algebra - G. Strang (alongside the mit ocw course also by gilbert strang)
an introduction to linear algebra, matrices have never been so pretty
2)Mathematical Methods in the Physical Sciences - M Baos
a good all round welcome to a lot of neat things, integral transformations, differential equations, orthogonal functions, the works
3)How to prove it - A Structured Approach - Daniel J. Velleman
this is the single best 'intro to writing/reading proofs' book I have ever seen
4)Differential Forms - A Compliment to Vector Calculus - S. Weintraub
differential forms are pretty handy, the goal of the book is to prove stokes theorem
5)Introduction to Tensor Calculus and Continuum Mechanics - J. H. Heinbockel (this one is free online)
this is the single best book on tensors I have ever seen.. seriously.. you actually get to understand what tensors are and what they do
6)Baby Rudin aka Principles of Mathematical Analysis - Walter Rudin
introduction to real analysis (this is why you read how to prove it )
7)Modern Algebra with Applications - Gilbert Nicholson
a nice little coverage of modern algebra, fields, rings, quotient groups, they're all here
8)Advanced Linear Algebra - Steven Roman
don't be scared by the fact it says graduate texts in mathematics, you'll be able to understand it from what you know by now
Good luck!