If you like to drive around on holiday, I recommend getting a car for 3 days and just exploring. The interior of the island can be stunning (red soil, olive/fig/almond trees, etc.) A few nice places for a pit stop are Santa Gertrudis (central, not too far from Sta Eulalia). There's a great bar called Bar Costa - go in and check out the artwork and hams hanging all over the ceiling. Also try the bocadillas (sandwiches).
Santa Anita also has a few nice bars/places to eat. Bar Anita is popular as the old "hippy" hangout, but if that's full then we found a lovely place just round the corner - great salads for lunch.
Also, check out some of the Ibiza websites which mention all the different beaches, and find a nice beach for a day. There are some stunning little coves, which you need a car to get to. Be warned that some of the roads are a bit hairy! Cala D'Hort is lovely, and the drive over the mountain from St Josep is also very scenic. You'll see amazing views of San An.
Oops, just noticed you asked about day trips for a family... I would still recommend all of the above, but depends what your children like to do/see and how old they are. Driving about may not suit them, although most places only take an hour or so to get to.