Creating alternate political maps with real continent boundaries

[Daelin]

For real-world maps, pretty much in any style you want, you can try snazzymaps.com. Tons of customization options. You can remove all political borders, cities, infrastructure, and essentially have Google Maps with just land and water and terrain in any colors you'd like.
Then you could use editing software to draw on top of those, make your own borders, labels, artwork, what have you. For this, Photoshop is the expensive option; alternatively, GIMP is a totally free image editing application that'll serve most non-professional needs.
Now those are your tools. They might take some time to get familiar with. As for making maps that actually look halfway decent, welcome to Earth. Most of us try our best here.