I wouldn't be able to offer detailed suggestions since I use them a little infrequently, but a few of what I've used have already been suggested here. Namely, Paint.net and Krita.

There's also some Wikipedia articles and guides out there with some decent info too. Here's an example you can look at, and then you can take a closer look at any programs that seem like a good fit.

And if you're looking for vector-based programs, I'm sure there's another article that should cover that.