Digicams are one of my favorite subjects, and I could go on quite a bit...
Sounds like you want a digicam to fill the role of a 'snapshot' camera, if you have a 35mm SLR then this makes a lot of sense.
I'd say this: Get a 2 or 3mpixel camera: 2mp will print up to 5x7 and 3mp 8x10 without much artifacts, if the image is saved in a lossless format.
Get a camera that can save uncompressed pictures. Many low end ones will only save .jpg's, but .jpg is a lossy format, think of it as the .mp3 of the visual world. Some mid-range cameras will have a lossless compressed format, and that is what you will want.
Small and light. The advantage of a digicam is that you can take great gobs of pictures. The more you carry it the more you will take, the better results you will have. I traded more features for more size (Cannon Powershot S30) and wish I had not. Not that the Powershot is big, but it needs a belt pack, and I wish it was more shirt pocket sized. You really want something that you can pop in your pocket and snap away like crazy.
Zoom. Must be a 3x zoom, you want to get to the 35mm equivalent of a 110 focal length. I forget who said it, but some famous, unfortunate, combat photographer said something like: "If your shots are not good GET CLOSER."
Brand. Cannon and Nikon have great software and community support, and that really is worth something because the OEM software tends to be crappy. Sony, Olympus, Pentax and Fuji less so, but some.
Speed. Many cameras suffer from lotsa lag in shutter release, focus, shot-to-shot cycle times.
Memory: At least two 128mb cards.
Battery: AA is nice because you can buy them in a pinch, but might have to go with a propriatary format to get a small camera.
Software. You need at least two applications: One to view, sort and archive, and one to edit. On the view sort, sort, archive front the lead contenders are Thumbs Plus and ASDsee. I chose Thumbs Plus for its better, less buggy support of Cannon's compressed, lossless format. Check them out.
Editing. There are a bunch of options, but after trying them all I'd say your best bet is to go with Adobe Photoshop Elements. Because: Anything you read about how to work with your pictures will be geared to Photoshop, and with any other program the help just won't be there, or you will be trying to translate the instructions into the other software. It just is not worth it. Plus there are tons of plugins for it that may or may not work with "photoshop compatible" offerings.
Sites: Here are some great digital photography and camera review sites:
http://www.steves-digicams.com/
http://www.dcresource.com/
http://www.dpreview.com/
http://www.imaging-resource.com/
http://www.megapixel.net/html/issueindex.html