Bradbury Mountain is a great little peak for a few reasons - namely it is very family oriented. It is very clean. Easy access for parking & even camping. There are nominal entrance fees - check out http://bradburymountain.com/.
Upon entrance you'll see a well maintain children's play area including slides, swings and picnic tables. You can hold events there or just an impromptu cookout.
There are several routes up to the peak. We usually take the steepest route for the best work out - the Summit Trail. It's only 0.3 miles up and is beautiful all the way. Great spot for family photos. After a snack at the top and some bird watching (this is a popular spot for bird watchers - lots of hawks & falcons spotted up there) and points of interest locating (the ocean - where is what town from the peak) we head down the Northern Loop which is easy going down - great for expending your kids remaining energy in a race - 1.0 mile down & out to the parking lot.
This is a great place for a picnic on the peak or a cookout at the trailhead - even if you don't intend to summit.