You can use any free paint tool actually (like Gimp). For HD graphics I suggest you pick some references and use vector-art using the tutorials and software blinkok has already shown you in his link.
Although if I were you I'd start doing pixel art and studying color theory. Pixel art is really fast to produce and you can get really nice results by choosing the right colours.
Check this
http://drakeor.com/wp-content/uploads/2014/01/Color.png out.
Also, try what people are calling "micro pixel art", which basically using resolutions so low that the traditional drawing rules kinda break-up (lightning stuff still applies) so you can make very fancy and original graphics just by trying a bit, just like the game "super brothers" for example.
EDIT: also I forgot, this
http://www.aseprite.org/ tool is an amazing tool for pixel art and it has a very powerful rotating tool wich will simplify your animations a lot if you are a beginner