Mathematica can be difficult to learn, but there is one area where it has very important advantage: the documentation is alive. The documentation is there not just to be read, you can modify the examples and run them in place just to see what happens. If you find something interesting, copy and paste the code into a scratch notepad of your own.

There's not an easy answer Joe because it's really not possible to learn Mathematica in a hurry. I always recommend that high school students interested in a technical career learn Mathematica during their vacations, or after school, or as a hobby. Then they'll have a running start when they get to an actual mathematical course and can think more about the math. But I think few do this.

What you can do is use the Documentation Center. Look at the "How to" tutorials. Study the Help pages on the Core Language and especially Functional Programming. Also learn about using Rules. Ask questions on this site using particular SHORT examples. Post the code by copying from your notebook into the panel, selecting and using the Code Sample button on the far left above the panel. Whatever you post should contain all the definitions needed so that it will evaluate, or illustrate the problem. So, study and get help.

Learning by plunging into a difficult math problem is a poor method. Try simple problems first where you understand the math. Learn how to translate them into Mathematica. Don't think of Mathematica so much as a conventional programming language. Think of it more as writing definitions, rules and routines for performing various transformations of mathematical objects. Longer problems can usually be broken into smaller routines.

I think it's useful to learn how to use the Sectional organization of notebooks. You can then keep different topics or cases in different sections. If you're having trouble with one of them you can mark it "Try 1"., copy the entire section and try a different approach or variation.