These days you can get some very good digital compacts - some of them are, depending on what you want from them, all but comparable to basic SLRs.
Here's a few pointers, but they're only my opinion (being a devoted digital SLR user)...
More megapixels are good, but don't get too hung up on it. The higher the MP count, the better quality your images and the bigger you will be able to print them without them starting to distort, but also the higher the MP count, the higher the price - and beyond a certain point it only makes a real difference for professional printing purposes, and if you wanted professional level pictures you'd be buying an SLR.
More MP does mean bigger image file sizes, so the more MP, the fewer images you can get on a given memory card.
You will always need to have a couple of spare memory cards - just like any other computer disk, they can get corrupted and stop working. This is also a reason that you should download the images from your camera or memory card to your computer regularly, and archive them if you don't want to lose them.
Digital cameras can be quite heavy battery-users, especially if you take pictures by looking at the screen on the back of the camera, rather than the traditional viewfinder. You'll always need to carry spare batteries because there's nothing more frustrating than having the batteries pack up when you're trying to take pictures! That said, by using the screen on the back rather than the viewfinder, you won't have problems with chopping people's heads off in pictures as the image on the screen is the picture you'll take, rather than looking through the viewfinder which is positioned slightly above the lens so not quite the same as the picture you'll take.
Digital zooms are usually very good quality now, but if you use an in-camera digital zoom to get closer to whatever you're taking pictures of - rather than changing the lens on an SLR - you will lose resolution in your final image file. So, you will have a zoomed-in image, but it will be of lesser quality than pictures taken without the zoom, and you will not be able to produce such large prints at such good quality. Hence, more MP = better zoomed in images, but more zoom = lesser quality pictures.
There is loads of choice for image-manipulation software available, the market-leader is of course Photoshop, which is very expensive but incredibly powerful. However, in-computer manipulation is not and should not be a substitute for taking a good photograph in the first place. If all you will ever want to do is maybe crop a little (I'm semi-pro and this is all I ever aim to do with a picture. If I have to do serious editing, the picture's not good enough), you don't need Photoshop, it's a waste of money.
Don't get hung up on cameras that do in-camera image manipulation. If you get one, switch the function off and do the image manipulation on your computer. In-camera manipulation is often not generally very good quality and will distort your basic image. If you save it to your computer first, you can save a copy, do what you want with the image, most likely do it better than your camera would have, and still have a copy of the original image if you make a mess of the editing.
If all you want is a point-and-shoot compact camera, the choice is vast. The more functions and flexibility you want, the more you're going to have to pay. What you pay the extra for with SLRs is functions, flexibility and image quality (regardless of the pixel count, a physically bigger sensor gives better pictures). Even if you leave an SLR on 'automatic' mode it will still usually take technically better pictures than compacts.
You can get perfectly usable compacts on ebay very cheaply if you're not picky about brand. The top brands are of course Canon and Nikon, with companies like Pentax and Olympus coming in behind. They do make the best cameras (technically, anyway) but you will also pay a premium for their brand names.
It's worth buying some photography magazines and reading the reviews to find out what models are just coming into the market and how good they are.
I think the best way to find a camera you like is to go to your local camera shop and try some, even if you don't buy the camera from them. These days, digital compacts are getting smaller and thinner, which is great if you've got small hands and the camera won't be getting many (anticipated!) knocks. On the other hand, if you've got larger hands, the smaller models probably won't be physically comfortable to use. There's no substitute for having a camera in your hands and finding out if you can hold it comfortably and operate the controls easily and see through the viewfinder comfortably. If it's difficult, you probably won't use it much, which is both a shame and a waste of money.
Topic: Anyone into digital photography