Bike ride on the old railway line between Burley and Brokenhurst stopping for cake at the old station tea rooms
Picnic, bikes, kites on Wilverley plain near the same tea rooms
Geocaching near the landlocked and hidden submarine pen and WW2 bombing ranges by Hampton Ridge (park on Roger Penny Way and walk/ride to the ridge
Walk/ride round Boldre Drive and Rhinefield Drive and look at Giant Redwoods and see the deer at the end (luck required, but not much!)
Elling Tidal Mill (I’ve not been, but it’s in a nice area and it gets good reviews)
Moors Valley country park. Little bit more organised, but good kids play area and trails and a gentle single track suitable for the family or a quick blast on the bike
HTH’s