Take Omag's advice & go & try some out at your local camera shop. Don't rush into it.
That would be Charlottesdad's advice... :P

But I second it. My 2 cents? First and for all, set a budget. Look what's available in that range.
option 1) The bridgecamera ( or superzoom): Small, a big optical zoom ( 10 - 12 - 18x), you can learn to control all the settings as on a dSLR and you don't have to buy additional lenses. Personally I have a Panasonic Lumix FZ5. Look at the newer models like the FZ8 and FZ18. Fuji is also known for great value for money. They especially have a low noise advantage. Click the link in my signature on what type of shots you might get with a bridge.
option 2) dSLR. Far superior in quality
IF you are dedicated to invest some money and time in the hobby. I recently bought my Canon 400D and I'm still figuring out all the controls and how it behaves. Don't be fooled by the low prices on the Canon 350 or Nikon D40. Though they are good camera's, you'll still have to buy additional lenses. My set-up ( 400D + kitlens EF-S 18 - 55 mm + Canon 70 - 300 mm IS) will have cost me about