Like everyone else, I'd recommend a D-SLR. You'll get better auto-focus and as you learn more about photography, you'll be able grow into the camera as opposed to growing out of one.
Because you're specifically interested in pet portraiture, I'd recommend that you get a short telephoto zoom lens with the camera. They'll give you more room to work with your subject and they generally render a more pleasing result for portraits in general (of both the pet and person variety). Typically, short telephoto lenses come in the range of a 50-200mm (or so) zoom lens for most manufacturers.
Some other things to consider:
1) Get an extra battery. It's always a pain if you run out of power and you want to keep shooting.
2) Get an extra memory card. It's always a pain if you see a good potential shot but you have no room on your card and need to start deleting photos to make room - which takes time, and pets don't sit still for too long.
3) When the budget allows, get an external flash. They're infinately superior to the built-in flashes on most cameras. They're less prone to causing red-eye, are a lot stronger than the built-in flash and offer more control over the output.
4) When the budget allows, get a tripod. They're a pain to tote around, but mine has been the difference between a reasonably sharp image and a truely crisp, sharp image. They're worth the investment and they open up the potential for different types of photography, such as night-time exposures and IR photography.
**Note: Normally I'd recommend the tripod before the flash, but because you're looking at portraits specifically, the flash seems a little more appropriate**
Hope this helps!
I'm going to start saving money now but would love some advice on which type of cameras to look into...
