Verified answer

You'll be better off with an entry-level SLR like the Canon EOS 100D/Rebel SL1, or the EOS 700D. Compacts or P&S cameras just can't take fast action. And for portraits, you'll need a proper fast prime lens (like a nifty 50mm f/1.8) in order to get the bokeh (blurred background).

For fast action photos you really need a camera which can be used at eye-level, not held at arms length looking at the screen on the back of the camera. Thus you should look at 'bridge' cameras, which have eye-level viewfinders, or possibly even DSLR cameras, but they are more expensive. You can get good bridge cameras under $350,but don't go too cheap. Both those sorts of cameras are also good for portraits.

At that sort of money I would be looking at a second hand DSLR like maybe a Nikon D80 and for portraits a 50mm f1.8 AF-D lens. Which action photography lens will depend on how close you can get to the subject, but you may need to save up for longer to get the right lens for that.

Age does not matter. If you can't take great portraits and action with your P510 now, you won't either with a better camera so stop bugging your parents and concentrate more on your schoolwork.

Unfortunately, you can't afford anything better than you actually have - learn to use it.

I don't usually recommend the P510 because, although an excellent camera, it's a bit on the expensive side. While not ideal for portraiture, it isn't bad at all.

Sorry but you won't find a dslr for your budget. They start at $500 and go up.

There are several models of controllable bridge cameras for this money. Fuji makes a pretty good line of bridge cameras... why not shop them.