Are there specific pieces that you want to be able to re-create? I can give more specific advice if you have one you want to point to.

For vector stuff (crisp lines, shape stuff), use Inkscape. The tools to start with are the selector tool (move shapes), node tool (edit shapes), and pen tool (the most powerful shape too, it'll take a while to learn this one). The dialogs to start with are "fill and stroke" (control-shift-f) and "align and distribute" (control-shift-a). The best way to learn these is to piddle around with them for a while, then go read the docs on these tools, then go piddle around some more.

For photo manipulation (editing or combining existing photos), use Gimp. The things to start with are layers, copy/paste, unified transform, "add layer mask" (in the layer menu), the brush tool (for editing layer masks, actually drawing in Gimp is not a good use of the program (also this is what you should use whenever you think you want the lasso tool)), and colors->curves. This is a pretty advanced way to use the tool and overkill for a lot of things, but it'll also work on the vast majority of photo manipulation jobs you might want to do. The best way to learn these is the same - piddle around, read the docs on the tools, piddle around some more - except it'll take more loops back and forth to the docs.

For painting, use Krita. I don't have anything useful to say about how to learn to use this one.