There is a Sealife centre in Weymouth, about 15 minutes walk from the town centre. You can also visit Portland which is not too far away.
If you have a car, you could go to Monkey World or the Tank Museum both near Wool. A little further away is Corfe Castle and the Swanage Railway. There is also the deserted village from the Second World War (name escapes me, but it's near Wool), which is quite interesting and somewhere different to walk around.
Dorchester is not too far away, but admittedly there's just shops there. There is a Maize Maze nearby, towards Abbotsbury I think.
Back into Wemouth and there is Brewer's Quay which has some local independent shops and the museam which is quite good. Then there is Nothe Fort which is worth a visit.
Given it's the summer season, I'm sure Joe Pasquale is back on at the Pavillion again!
Finally, if you're feeling adventurous, how about a day trip to the Channel Islands on the Condor.....!
Not sure for whom/ages you're asking for but it's a few suggestions anyway. I'll be in Weymouth myself in June (not that that is an attraction!).