M.J.
I just went through this research process about six months ago. Same situation. Not wanting to buy a DSLR due to expense but wanting to upgrade from our basic point and shoot, which wasn't capturing our kids in action.
I did a lot of reading online and finally went to a camera shop, which confirmed the Canon SX-30 IS was a good fit for our needs. We've had it about six months now and love it! It does take some getting used to carrying a heavy camera around, but we're getting all the shots we missed before. Have captured our kids mid-air jumping in the pool, running shots, etc. Nice, clear photos. And the thing I like most is that it's easy to figure out. I just don't have time to learn fancy electronics and this one isn't very complex.