There's been some good general advice, especially the post about the non-camera specific stuff like storage, cards and batteries. You have to figure out what is important to you. Size? Image quality? Features? Price? Build quality? Truly can't have it all. And it's not really a money thing, no one has yet made the perfect point and shoot.
OK. Now I don't want to get into a religious discussion about cameras but some things to stay away from. Proprietary memory cards (meaning nix to Sony), noisy imaging chips (generally many of the Panasonics), and proprietary Li-Ion batteries (everyone, especially as you get to the ultra-compacts). SD (Secure digital) is a great standard. And anything with AA's are great too (but a little bulkier) since you can get spares anywhere - though rechargeables are nice. And cameras using AA's generally have better flash recycle times. But they're a little bulkier and heavier.
At your price point though, pretty much anything you pick will be roughly equivalent to everything else. Just not that much differentiation at the low end. That's still in the price range of a throw-away commodity from the manufacturers viewpoint. A major failure of the screen, lens or chip and you'll find it's cheaper to buy a new cam rather than pay for a repair in most cases.
You should use the buying guide at DPReview
http://www.dpreview.com/reviews/compare.asp. For ~$300, two of the top choices would probably be a Canon SD800IS or A640. The 800 has image stabilization and a nice 28mm wide angle end. The 640 is pretty much the champ for image quality under $300 (although the Fuji F30 does better in low light).
The difference is the sensor size (see
http://www.dpreview.com/learn/?/key=sensor sizes). Most cams under this price will have a 1/2.5" chip. And most will all be made by the same manufacturer so it's camera features to concentrate on if you go that route. The A640 though, has a 1/1.8 chip. Math-wise, that's about a 54% larger imaging surface. Size matters here. But the 640 doesn't have IS. Although at 10MP, and very decent high ISO performance, low light pictures for a 4x6 or 5x7 will be fine.
For megazooms and simple short movies, I would also recommend the Canon S3. Has IS and, on sale, can be had for slightly under $300. The S5 is coming out in July (not worth the upgrade) so prices should be falling a little soon to clear stock.
Good luck choosing! You'll need it:lol: . Some of my 'net buddies spend more time trying to figure out which under-$500 camera to buy than they do when deciding on a new car.:ROFLOL: Seriously.