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I recently gained acesss to adobe products. Before that I used gimp and inkscape for years. There is no reason that anyone couldn't replicate what I did in those programs.
A tablet is only a tool. Just out of curiosity what software are you using now?
I was in a situation, probably, where you are. Before worrying about making logos or anything else you should check out the following link. Check out everything in the post. It develop the fundamentals. Even if you feel like you have them down a review is always nice.
Awesome. Just be sure to checkout that link. if you want other resources like those I have tons of bookmarks(some of which I need to read/review myself)