The best advice I have ever gotten:
Do the exercises without looking at the suggested solutions. This will force you to think critically. In the beginning this will probably be very time consuming, but do not give up, because after a while the payoff will be huge.
If you really get stuck, discuss the problem with someone else. In this way, you will have to identify the reason(s) why you are stuck. (you could for instance ask for hints at this site).
I would also advice you to read (a lot) about how professional mathematicans approach mathematics. Have a look at the following pages:
Terence Tao's blog contains a section on Career Advice, which I really appreciate. Terence Tao and Timothy Gowers have several informal discussions about mathematical topics on their homepages. I really benefited a lot from them.
Also, on the philsophical level, do not get too obsessed with trying to understand the intrinsic meaning of mathematical notions. Mathematical objects are important because of what they do, not because of what they are.
To quote what John Von Nuemann once said:
"Young man, in mathematics you don't understand things. You just get used to them"
(A reply to Felix Smith who said he was afraid he did not understand the method of characteristics.)