Probably a prosumer model which is between a compact and a DSLR would be a good choice. Still lots of room to grow as a photographer and learn manual settings but also fully automatic. Most have a continous or burst shooting mode that will let you take several shots in a row and shutter lag is less than in compacts.
Brand is a different thing altogether - Im a Nion fan but ist best to fins a camera with the features you want rather than a particular brand if you dont already have an allegiance.
Compare features by going to
http://www.dpreview.com/ and viewing the newest models in each section. Look for features such as continous mode speeds, shutter lag and perhaps image stabilisation - which makes it easy to take photos of the kids indoors, and then you can do a side by side comparison also.
HTH