I only have an old Kodak 3.2MP digital camera to work with...
First of all, forget what camera you have, it's completely irrelevant and in no way makes it harder or easier to get an "arty" photo. A good photographer with a camera phone will take better pictures than someone very average with a DSLR.
Secondly, as a starting point you can't go wrong with the
rule of thirds, just knowing that will improve a photo immediately, as most people automatically put the subject in the middle of the frame.
Another thing that can help, get down low. 90% of photos are taken from about 5-6ft, getting down near the floor for a different perspective can really make a difference.
Most importantly though just experiment, tilt the camera a bit, get in close to subjects and photograph the details instead of the whole thing etc etc. Everybodys "arty" shots are different because we all have our own unique style.