You basically have two problems with Point and shoot digital cams:
1) The reach: a lot of the lower budget cams only have a 3x zoom, equivalent to something like 28-84mm or something. This is generally too short to take anything "correct" unless you are sitting at the very front row...
You can now find more and more prosumer point and shoot cams that have a 10 or 12x zoom. I am no specialist, but some of those may be in your price range...
2) The sensivity (comparable to ISO): Point and shoot camera have generally a low sensivity, somewhere around 50 to 400 ISO. Also the quality of the pictures at 200 or 400 ISO is often awful.
I won't enter into detail, but this basically makes it such that you won't be able to take any good picture indoor basically (because your shutter speed will be too long).
Outdoor, you can have enough light, but you have to ensure that the shutter speed is high enough. You will need something around 1/200s or faster to have some good motion freeze...
Hope it helps a bit.
There would be tons of things to say, but check www.dpreview.com
, great website on digital photo with some great forums as well
PS: these are just a few pics i took with my D20 last year in Zurich. Now of course it is a DSLR and not a p&s http://www.wtaworld.com/showthread.php?p=6501325