Depends on a couple of things. Do you own some portable devices like MP3 players or perhaps another digicam? It's important because they use memory cards and you can perhaps recycle whatever memory card format they use with your new camera and save some money that way.
It also depends on what you will take pictures of. If it is just people, flowers and indoor shots, then any general use camera that is for you.
Off the top of my head, you can consider the Canon PowerShot A70. It's a 3-megapixel camera, is very well liked and it has a whole host of settings and manual controls, just in case you grow tired of its automatic mode at some point and decide to learn more about photography. It's possible you will find its successor the A75, but I insist you look for the A70, it's practically the same camera and you can get it for at least a $100 cheaper.
There's the Nikon Coolpix 3200. It's a 3-megapixel camera with lots of features. It's nice and small. Heard some great things about it. You can find it around $300, I believe.
Despite my hatred for Sony, they do make some nice digicams. The Sony Cyber-shot DSC-P72. It's 3 megapixels, easy to use and small. Might be for you. The 4 megapixel DSC-P43 could also be an option for you (costs a bit over $300):
Fuji makes some nice low-priced camera. The FujiFilm FinePix A205 is one of them. The Finepix A330 and A340 could be for you as well. You get a bit more megapixels with those for some extra $.
However, if you're going to take pictures of landmarks, landscapes and animals on farway tree branches, you will need either more megapixels to get those details (take big detailed pictures and then crop the part you want) and/or more optical zoom to actually get those shots (you can't afford most long-zoom digicams, but you can get a bit more zoom with certain models). And don't listen to sales clerk, a
digital zoom is useless.
The Minolta DiMAGE S414 is an interesting model. It offers a lot of value with its 4 megapixel and 4x zoom. It is a bit like the Canon A70 with an easy interface but with manual options underneath in case you want to be a more serious photographer. You can find it a bit over $300.
Find the Kodak EasyShare DX6340. It's a 3-megapixel camera with a 4X optical zoom (the industry standard is 3x optical zoom). The Easyshare cameras are among the easiest to use on the market, but this model probably all you want and it is among the highest rated in the Kodal models. You should find it for a little bit over $300.
Alternatively, Kodak has this low-priced 5-megapixel model. It's pretty like the other Kodak model except you trade the extra optical zoom for more megapixels. You can find it under $350.
Then there's the FujiFilm FinePix 3800 Zoom. This is a 3-megapixel camera with a 6X optical zoom. I think it sells close $400 but if you shop around or find it used, it could be possible to afford it.
The same goes for the Minolta DiMAGE Z1, a similar camera except you get 10X optical zoom. The price can be tricky for you (around $400 as well), but if you're lucky, you might find it much cheaper.
Anyway, that's what I can think of for now. Please let us know what you have chosen.