Keith, in this post,
http://www.thephotoforum.com/forum/photography-beginners-forum/275338-2-new-spring-photos.html, referenced the PPA's (Professional Photographers of America) criteria for a Merit Image. This is as good a place to start as any. The very first item on that list was "impact", or the emotional effect on the viewer. If you have a P&S, you don't have much control over the technical aspects, but you do have control over things such as your viewing angle, your framing, your choice of light, the timing for taking an image, the subject matter, and the environment/context in which that subject is shot.
In discussing impact, you can also touch upon things such as simplicity, focus on one message (or theme), use of symmetry or balance, and creating s sympathetic link between the subject in the image, and the viewer. The use of isolation is a form of simplicity, allowing the subject to be isolated from its context so that we can appreciate it on its own merits. Of course, the techniques available to a good photographer with the right tools (a DSLR, for instance), will make the task easier, but even a P&S (or a cell phone camera for that matter) should be able to capture images of interest.
Personally, I would focus on cultivating a sense of "seeing". This book is an excellent guide for exploring this subject.
Freeman Patterson : Photography and the Art of Seeing..
Another approach is to try something like this:
The Beginners' Lens
And most of all, have fun!