I'm guessing your current camera is a compact pocket-style cam (You mentioned 'Cyber Shot'...FYI that's actually the name of Sony's entire digital compact camera line, so there are dozens of models with the name Cyber Shot on them - the model of camera is usually a letter and number combination, such as 'S500' 'W300' 'H5' T100', etc.).
If this is the case, a nice move up for you would be to look at an 'ultrazoom' model of camera. These are usually more responsive when you press the shutter, focus faster, and have a bit more advanced anti-shake, focus, and metering modes that can help improve photos in 'Auto' mode. They are not pocketable, but are still significantly smaller and lighter than a DSLR camera (the kind that take external lenses).
Check out Sony's H50, Canon's S5 IS, Olympus' SP570, Nikon's P80, or Panasonic's FZ28, to name a few. Most of these are under $500, take great 'auto' shots for those who just want to point and shoot, have massive zoom ranges that can shoot subjects right in front of you to distant objects on the horizon, and have advanced features and manual controls in case you ever decide to experiment or learn more about photography (room to grow, if you so choose).
DSLR's are superior cameras overall, in their abilities with low light and action shooting especially...but typically are more expensive, much larger and heavier, come with more of a learning curve, and tend to work best when you have a modicum of knowledge about photography. I wouldn't recommend you go that direction at this time, though others may suggest them (and yes, they too have 'auto' modes, but the functionality is quite different and buying a DSLR to use in Auto mode tends to be a bit of overkill - as you are not coming anywhere near getting the most from the camera, and could likely do as well and easier with an ultrazoom).
