Camera
There seems to be lots of helpful tips on this already but here's my advice if it's any help.
1. Size Matters:
Almost any digital camera from £50 up nowadays will take great pictures, so the most important thing is that you can carry it comfortably, put it in your bag, or purse, pocket, etc, OR if you're happy to carry it in a dedicated bag, that's cool too as it allows you a stronger camera. There's some really good cameras for under $100 that you can use underwater, that are shock and dust proof, etc, and these are handy as you don't have to worry too hard about them, and if you lose it, it's not the end of the world.
2. Megapixels don't matter, Price does:
Don't agonize over a 12.1MP camera v an 8MP camera or any other alternatives. Look at the price. Are the cameras similar? Is one much cheaper? Go for the cheaper one, maybe last year's model they've not sold!
ONE thing re newer models is this, they REALLY seem to improve the image stabilisation every time and that makes a big difference, my advice is to try them out, bring an SD card and look at the photos later on a computer or printed. The camera records the active settings into the photo file so you can see which one you used for which photo (EXIF).
3. What type of photos do you want to take? Time of Day, Indoor/Outdoor? Near/Far?
This is something people forget about when buying cameras but it's really important. I have a Leica V-Lux which I ADORE, but it's rubbish at taking indoor photos in low light. It's amazing at everything else. Will you be using the flash? Most cameras have rubbishy built in flashes but many have 'hot shoes' that you can plug a professional flash into.
The max ISO ability of the camera (e.g. max of 1600 or 3200, etc) has a bearing on low-light photos and it's worth trying out the camera you want to buy in low light to see how it performs at high ISOs. There is often a graininess to photos at high ISOs but some cameras are significantly better than others in this regard.
At the opposite end of the scale, taking photos in bright sunlight or where there's backlighting, it's worth getting a camera that you can put a polariser on as it can block out some of the sun's rays and reduce the light coming into the camera.
If you're going to be doing lots of close-ups from close-by, you DON'T need to get a camera that can zoom 60 X.
If you're going to be taking close-ups of things that are far-away, you DO need a camera with a good zoom, but also one with good 'auto image stabilisation' as at that level of zoom, your hand's natural shakiness really impacts your photos!
4. Making the Choice:
You've mentioned a budget under $500 which is cool, you can still get a great camera, just maybe don't go for some entry-level SLR (with the exception of the Lumix G-series).
There's loads of other factors to consider
Personally, I always recommend Panasonic's fantastic Lumix range of cameras. They use Leica lenses and they are really good cameras. The G2 was launched recently and although I've not gone out shooting with it, it seems to be a really good option as it allows you to expand your camera as you expand your needs and your skills. You're buying into what's called a "system camera" [system], which is like with consumer and pro SLRs (Nikon/Canon being the really popular ones), you buy the lenses, bodies, battery grips, bit-by-bit as you work your way up the foodchain.
5. The Canon / Nikon Debate: Ignore it!
People might tell you that's a good idea to buy into your system of choice but my own opinion is this. There is ZERO point spending a thousand dollars on a bottom-end Canon EOS or Nikon SLR with a crappy consumer grade plasticy lens.
If you're serious but nervous, start off with something with good manual controls, but also good automatic controls like the Lumix FZ models or maybe a second hand canon 500D or whatever's good that's on sale, check dpreview to find out about a model you see on offer. This allows you to understand how these manual controls affect your photo and then if you still want to, you can buy a PRO Nikon or Canon.
Why not just buy into the system straight away?
Cuz let's be realistic. You're not going to use a crap lens that you got free with your 1000D (entry-level Canon) on your 1D, 5D or 7D body, and likewise, you're not going to use an amazing lens on a crappy body because it's not worth it, so having loads of lenses might seem great, but it's of no more use than having a great small camera and a great big camera, and I'd argue that this is a much better solution. If you're looking for a 'big' camera, I love LEICAs, but also have experience shooting with more mainstream cameras and LOVE the Canon 7D (it's a really good balance between affordability, spec, weight, ease-of-use, etc), and also the Hasselblad H3D and H4Ds. Hasselblad and Leica are shockingly easy to use. Instead of masses of tiny buttons, all the controls are easy to find and simple.
Ok, so what are the great small cameras, cuz this is what you're looking for really:
Well like I said earlier, the Panasonic Lumix G2 is great, but also the G1 and G10, the FZ and TZ range are great as well. TZ's are more automatic with less manual control but still a great camera and great shots from them.
Canon G-series (G9/G10, etc) are fantastic cameras too, and I would highly recommend them, some of them are better for low-light photography than the Panasonics.
6. Save on the camera, splash on something else!
There are lots of other good cameras as well, and it all depends on budget. You can always buy a cheaper camera and then buy a lovely bag or a better hotel room.
My advice is, if you're comfortable buying second-hand, do that! You get something so much better for so much less!
Just research first, and if you have any questions about a model, look on DPreview or feel free to ask!
Oh, final note! Loads of mobile phones now have fantastic cameras and it's worth considering maybe putting the money there instead. The iPhone 3G S and 4 have touch to focus which is really easy and takes great snaps, and the blackberrys and some androids let you use the LED light for filming... The newer blackberries (such as 9700) take great photos. The Curve takes ok photos and the old bold (9000) takes RUBBISH ones. :-)