Dogs are welcome on the beach at the bottom of the road (450 yards).  The ban on dogs does not start until about half a mile towards Bournemouth.
 
Go east, away from Bournemouth, and you can walk around Hengistbury Head on the beach, or climb over the top if feeling energetic.  Ether way at the end of your walk there is a cafe, or take the ferry across the Run to the pub on Mudeford Quay.
 
The New Forest is about a 20 minute drive away.  This is a huge area and whichever turning you take there will be lots of doggy friendly walks, plus numbers of cafes and pubs that welcome dogs.
 
For shopping, head either to Bournemouth or Christchurch - or try the Castlepoint shopping centre on Castle Lane (though your dog might be so impressed!).
 
Further afield, Poole make an enjoyable day out - check out Poole Quay.
For ship enthusiasts Southampton is worth a visit.
Car buffs will like the National Motor Museum at Beaulieu.
Further afield into rural Dorset there are any number of pretty villages, churches, pubs, garden centres, craft centres - and of course loads of walks for your dog(s).
 
You'll have to try really hard not to find an interesting way to spend your stay!