Digital Age

I don't like to use my online soap box for personal reasons, usually, and I'm aware it's going to make me a hypocrite since I just said a couple days ago not to buy anything for Christmas until next week, but I asked my Mom what she wants for Christmas, and she said:

As for myself, I am thinking I'd like you and your brother to look at digital cameras that are not too expensive or hard to operate, that have zoom lenses. I have been wanting one to try my luck with. When I get mobile, I may want to actually go places again!

So, unfortunately, I don't know anything about digital cameras. Never owned one, have little experience in using them. Help a sister out. Any advice/warnings/cameras you love would be much appreciated.