In June - Dover Island for climbing! It and other climbing are both very near Peggy's Cove, a popular tourist spot.
Around Halifax there are quite a few things to do and see, hiking/museums/drinking/old buildings/ocean and boat stuff/etc.
Other major tourist/driving sightseeing regions are probably:
- the "south shore" - mahone bay/lunenburg - UNESCO world heritage site.
- the fundy shore (both sides, cape split is a nice hike on the wolfville side, and 5 islands park/chignecto/parrsboro area is on the other)
- the annapolis valley, and
- cape breton (cabot trail and louisberg I suppose are the big ones here).
A fair bit harder to find climbing there than around Halifax though... For all of these a car is really the only option for transportation. Otherwise you can manage getting around Halifax on a bus, but you'd need to bum a ride out to some climbing spots (shouldn't be too hard though...
)
Good luck!