When taking abstract shots, you need to think more about patterns, colours and texture rather than concentrating on focusing and composition.
Having said that, someone may consider a shot of river rapids captured with a very long exposure as abstract thanks to the blur the water takes on and for this kind of shot, the rules of photography are still important.
If you're wanting to focus more on patterns and shapes then the key to capturing an image that works is to create an image out of something ordinary that you wouldn't normally see while still creating an appealing shot. Look for patterns on buildings, rippled reflections on water or see what a shallow depth-of-field can do when applied to a close-up shot of a flower. Textures work well too and they can also be applied to images in Photoshop to give them a different feel. Something as simple as panning the camera down a row of trees in autumn can produce lines of bold colour or try zooming your lens out during a long exposure and you'll add a zoom burst to your scene.
Working close-up with a macro lens will allow you to fill the frame, shooting one subject from various angles but still leaving enough mystery in the shot so the viewer has to think about your subject choice. This doesn't mean compact camera users are excluded from trying this technique as many do have excellent macro modes now meaning you can still get close to objects without a dedicated macro lens. The Pentax WG-3 compact is designed with macro work in-mind as it features LED Macro Lights around the circumference of the lens barrel for macro shooting and a Digital Microscope mode. These help the users capture a microscopic view on the world which means shots of a carpet, for example, can be turned into abstract work.
A very close-up shot of a carpet!
Colour is important for abstract work but don't make them too contrasty. This doesn't mean you can't have multiple colours in frame, you just have to ensure they all work well together before hitting the shutter button. If you want the shades to be more saturated try popping a polarising filter on your lens.