I take great BB pics with a canon 50D I bought used for $400 I think. I use a 70-200 lens thought that cost 1200.00
Pretty much any camera that takes good sports pics will take great family photos. If you desire to get very close to the action you will may want a camera on which you can change lenses. Some of these cameras with digital zoom are taking amazing photographs now days though. I have 3 Canon professional cameras EOS 1D, 5D, 50D I bought the 5D off craigslist for about half what a new camera sells for. Sky is the limit when you get into photography money wise.
One way is to find someone taking say basketball pics and ask them what they use. Most photographers are glad to share their knowledge about what cameras, lens, software they use with you.
At new prices, I am guessing you have either the 70-200 f/4 with IS, or the f/2.8 without.
My 'distance' setup is with the f/4 IS. The 'walking around' lens is the 17-55mm f/2.8 IS; great lens, but wish it had more long end. At least it's 2.8 all the way.
I keep a 1.4 telextender and the little 50mm f/1.8 in the bag just in case. Also a 430EX or 580EX strobe.
The great thing about the 50D is how little it will sell for, used. The not so great thing about my 50D is how little it will sell for, used. ≤}B^)