As someone who was taught the 'traditional' way of mathematics, can someone give a few pointers of de-programming myself from the traditional way that I was taught? (Although maybe it won't be so hard since I feel like I've forgotten quite a bit)
If you can code, Project Euler is really good. Particularly the problems past #100; those tend to concentrate more on mathematical insights than programming ones. It's what kindled my interest in math in high school.
Yes, but Project Euler merely gives you a stimulus to go out and research, and a test to apply your new skills. They are not so much a tool on their to acquire the skills.
E.g. learning about dynamic programming will simplify at least half of all Project Euler problems. But it would be rather harder than necessary to try and come up with all the generalities of dynamic programming just from personal attempts at solving Project Euler problems.
A problem that you are motivated to solve is a good idea. Because it provides clear motivation. If such a thing isn't forthcoming, maybe think of understanding the concepts as the problem to solve (which is pretty wishy-washy, but it is likely going to be a difficult task if it isn't providing any satisfaction).
Read `The Pleasure of Counting'. It's an excellent mathemical book for teenagers written by an actual mathematician and a gentle but real introduction to real mathematics. No dumbing down.
Someone already recommended this for calculus which looks pretty promising: http://www.gutenberg.org/files/33283/33283-pdf.pdf