I have no serial-output in my computer. Is that a problem?
Arduinos contain a USB to serial chip in them so there is no need for a serial output on your computer,
Would the an specific Arduino board be the best choice?
I would choose the standard 5V version you can get one from here:-
http://www.segor.de/Katalog/v2segor.shtml
How do i work with the motors?
http://www.thebox.myzen.co.uk/Workshop/Motors_1.html
I use linux. Is that a problem?