Depends on what you wanna do:
- Cartoon Illustration - I recommend PaintToolSai. I've been using it since I was in high school, and it's awesome. It has a huge user base on deviantArt, too.
- Logo Design - Illustrator / Photoshop. I recommend Illustrator because it creates everything as a vector, therefore, when you go to resize images, it won't lose pixel quality.
As for free ones, I believe there is one that's highly rated called "Gimp." I've personally never used it, but I know a lot of people who do. I think it does anything from photo manipulation to illustration. I could be wrong, though.
Before you do digital painting, you want to invest in a tablet. You could do it with a mouse, but it's just not going to be the same.
I have two, but I really only use one because it's more portable and convenient, and it's the Wacom Bamboo. You can get it off Amazon for about $70, as long as the prices haven't gone up over the years. It's smaller than computer paper and fits easily in your backpack, so if you have a laptop and need to work on art, you can easily store the Bamboo. The other one that I have is a Wacom Intuos, which is professional grade. It costs around $200. Really only get it if you're going to be a professional digital artist. Otherwise, Bamboo is just fine.