Last year, I researched and researched -- and became very confused. I bought a Fuji Finepix F20. I think it was about $125 or $150. I love it. After reading and listening to many sales pitches, I decided I didn't need half of what these cameras had. I have used very few of the extra features on this camera. I use it mainly to take family pix and pix to post here to get everyone's awesome help.
First decide what you want to use it for, then research based on that.
I am sure you will get many, many opinions. There are a lot of good ones out there - and as soon as you buy one, new ones are on the horizon! Good luck!

Start with a simple point and shoot with as many mega-pixels as you can afford (at least 8). My preference is either a Canon or Sony, but there are sure a lot to choose from.

Yes, decide what you want out of it first. I like to take closeups, so it's important to me to understand just how much zoom and detail I'm going to be able to get when looking at cameras. Some cameras have a zoom feature but they really don't get in very close.

I love my Sony camera - and did a lot of research before I bought it. A lot depends on what is important to you. I wanted something small I could put in my purse, a viewfinder for those really bright or very dark situations, great macros (for flower close ups), true colors (probably most important - my last camera always looked washed out), clarity, a large LCD screen, and an easy memory card/system. I ended up with a previous version of this (that I think was cheaper, too): Sony Cybershot

A sample from my yard :-)

This camera is not the best, however, at taking pictures quickly or in fast succession. You should really think how you will be using it! The link below was EXTREMELY helpful in researching all the options.

Here is a link that might be useful: Digital Camera Resource / Reviews

You can always buy a used camera off eBay for a starter and then upgrade, once you are comfortable with the digital process.

I recently bought my third Sony camera and consider them the best.

You can probably purchase a point and shoot Kodak for $100.

This is a feature from my camera that will most likely be useless. It the FishEye:
{{!gwi}}

My camera can change a background to Black & White, while keeping the subject in color:

Don't forget that most digitalis now have a video camera:

Here is a link that might be useful: Video of Easter Snow

Circuit City currently has a nice Samsung 7.2mp camera on sale for $75 with a free printer. Most people do not need all the bells and whistles, as others have pointed out, but the Samsung has a nice collection of features. You actually don't need super-high megapixels if you're not going to make big enlargements, do heavy cropping (like taking only 10% of a picture and blowing it up), or take really, really tight closeups, and very high resolution does chew up the camera's memory and hard-drive space... higher megapixels does not necessarily equate to a better camera.

I have a Kodak Easyshare camera and like it very much - it has stood up to quite a bit of abuse like getting dropped, left in the freezing or roasting car, etc. Five years old and operates just as well as the day it was purchased. (It skews a tad to the pink on occasion but if I weren't so dang lazy I could correct that easily, and it has been fixed in later models.) It is a 3.1MP camera and still takes perfectly presentable pictures for my needs - photo sharing here, family/pets snapshots, that sort of thing. Circuit City has the current 7MP model for $100 with a free printer; Best Buy has it for $100 with no printer. Some people really like having the photo printer, we never even used up the supplied package of paper that came with ours and ended up giving it away. If I need hardcopy prints I go to somewhere like CVS or Walgreens for them, since they frequently have coupons for free prints, and get good 4x6 and even 5x7s out of my sorry ole 3.1 even with moderate cropping. ;-)

A pawnshop is another place to get a cheap, used digital camera, and you'll be able to get your hands on it before you purchase to make sure it works.

I was trying to remember why I upgraded my old digital, and last mango reminded me- it was because I was concerned my kids would be grown up and off to college before the thing took the shot. Okay, that's a slight exaggeration, but when you're primarily taking pics of small children, its a big consideration, since they aren't going to be in the shot by the time the camera fires.

Mine was a canon power shot that's about 5years old, and now it belongs to my 5yo. Some other things to keep in mind are battery (some take AA, some take special order camera batteries), and storage card (what is it? an SD card or something?).

How large do you want the LCD screen on back? Is it important to you to have it be pink? Do you want to be able to slip it in the back pocket of your jeans? Optical zoom is better than digital zoom. Read lots of reviews.

Picassa is a nice free photo editing/organizing program that you can download to do things like take out the red eye, crop, email, convert to black and white, etc.

Viewfinder is one feature mine does not have that I would have liked. It makes it difficult out in the sun. As was mentioned above, you do not need the highest megapixels unless you are going to blow the pictures up to a very large size.
I do not have a printer and really don't want one. I don't think the quality is what you can get if you have them printed, and the paper is very expensive.
I upload my pictures to Walgreen's website, they print them, send me an email when they are done and I go pick them up. Easy, peasy and the quality is great!!

The reason I bought the camera with the higher mp is because of the video quality. We are going to the Mediterranean in June and I wanted a high quality video camera (in a little package) so that I could have some good footage to show my children.

I set the mp at a lower amount than the 12.1, so that I can get more photos on my 2G card. Even so, I tend to do a lot of cropping and enlarging, so the extra mp helps the picture quality.

My camera is small, has a large viewing screen, but also has a viewfinder. I felt I needed it to take more steady photos, but as it has turned out, I am taking many through the screen and they have come out fine (because of the anti-shaken mechanism inside). The camera actually has a computer inside and I can hear it working. Weird!

The camera was perfect for me! I watched the prices at Circuit City, and then jumped when the price went down. Plus, I ordered online and had an extra $$ coupon. It also came with the free printer (beware that you pay for the printer upfront, then you have to mail in a rebate)but I decided to return it after seeing the mammoth size of the thing!

If you can find a Sony or Cannon at a reasonable price, then go for it. I think the Kodak Easyshare is a perfect first time digital. I got my 82 year old Dad an Easyshare for Christmas one year. As simple as it is, he still has trouble with understanding the function.

I have enjoyed my Canon and my Sony before that. One thing to look at is shutter speed. I can't remember if that's what they call it on digital cameras, but I wanted it to take the pictures as quickly as possible (I think they talk about time to first shot and time between shots). There can be a lot of lag time on the digital cameras and that makes taking pictures of kids very difficult, especially in low light conditions. If you aren't taking pics of kids or other things that are moving, it probably doesn't matter that much. Just something to consider...

At first I missed my 35mm SLR, but now I don't use it at all and only use the digital.

I have 2 digital cams, love my tiny one I use it all the time however next one will have optical zoom, Target has one for around $89.00 also a Poloroid has all the features I want,and a 2.5" LCD screen too,6x digital and 3 x optical, so that will no doubt be it for me! there are lot's of good ones under $100