Bastiaan - welcome to the Guild!
There are many free or very cheap graphics programs available. A first question might be what style are you planning to map with? Different types of programs would excel at different results. Inkscape is free, and does nice clean vector graphics. http://inkscape.org/
The GIMP is also free http://www.gimp.org/ and is an excellent raster graphics tool. Both have tutorials in the Guild, and experts willing to help learn. And both are pretty much as good as the equivalent commercial products. There are dozens, if not hundreds of other choices - a thread devoted to the many possibilities will give you ideas. http://www.cartographersguild.com/sh...112#post195112
Another question might be what is your purpose? Maps for academic or professional use have some needs that fantasy or game maps do not, and vice versa.