Probably the best free image editing and drawing application you can use is GIMP (www.gimp.org). And if you want to go to the paid route, go with Adobe Photoshop — the current version is Photoshop CS5.
With both these software packages, it's easy to get started because you'll find hundreds of tutorials (even video tutorials) online for free. Note that if you have a tablet like the Samsung Galaxy Tab or Apple iPad, you can also download various drawing/doodling applications from their respective app stores, both free and paid.
Finally, you might want to consider purchasing a Wacom drawing tablet — you can use this in tandem with your software package of choice, and it'll give you enhanced precision as compared to just using a mouse.