Good composition – look around the frame before you press the shutter release. If there are things in the way, move yourself and/or the camera to exclude them.
Correct focus – there is nothing you can do with an out-of-focus shot.
Correct exposure – the camera will mostly get this right for you except in tricky lighting conditions.
“Cheap” P&S cameras will take decent shots but the more you pay for a camera, the more control you have over things like exposure & focusing and the better quality the lens. It’s just as easy to get composition wrong with a £3k dSLR as with a £100 compact, however.
A cheap/SH dSLR would give you a lot more to experiment with and action shots are more accessible because the camera will respond much more quickly (faster focus speed, minimal shutter lag, more frames-per-second etc.). Even if you can borrow one, give it a shot and it might just give you the “bug”.
Also worth trying to gain a basic understanding of how the main parameters affect the shot (aperture, ISO, shutter speed, exposure comp.).
Good luck!
SC