If you take a hire car, the Summit County towns of Dillon, Silverthorne and Frisco are good options for (relatively) lower cost condo accommodation and easy proximity to Breckenridge, A-Basin (beware the East Wall), Keystone and Copper Mountain. Vail and its expensive, impossibly great back bowls is about another half-hour (or more depending on traffic and road conditions) down the highway over Vail Pass. Try
here for openers. If you want to focus more on the kids' skiing, I'd say head for Keystone or Copper. Winter Park is nice but as
L-o-M said it's mostly Denver commuters, as is Loveland (which isn't really worth your time).
I've spent a ton of time skiing in the area and I second this advice. If you're trying to do the area on a budget, these towns are excellent choices. Much lower cost accommodations and convenient to multiple ski areas. Also plenty of shopping and eating within close range. You definitely need a car for the duration of the trip, though.
As for mountains that are good for both the parents and the kids, so a nice mix of skiable terrain and good all day ski schools, I recommend both Copper Mountain and Breckenridge, which are the two areas closest to Dillon/Silverthorne/Firsco.
Beware of the drive back into Denver if doing it on a Sunday afternoon. It can take hours, due to the volume of weekend skiers making their way back to the city.