I can't draw for sh*t, but Autodesk sketchbook is nice and free. By making a free account you get some extra stuff as well. The free version has a layer cap though.
All in all, GIMP is probably the best free one out there.
It's mostly just experience with illustrator. If you want to learn, try the free program Inkscape and watch some tutorials. This one seems pretty thorough.
After you know the basics, it's really as simple as creating shapes and aligning them properly.
Or, you could do what I did, download a trial of illustrator, and spend a month playing around.