The world, as they say, is your bivalve.
There are two country parks in the Stirling area, Plean and Mugdock, Plean is good for doggy walks but isn't that big. Look at
http://www.stirling.gov.uk/index/leisure/countryside/walks.htmMugdock I haven't been to for a while but it looked nice, and you could do the walk to the Whangie also, one of my favourites, fantastic views north for very little effort. The Ochills are good, Ben Cleuch etc., however, unless your dogs are trained to ignore sheep then you have to keep them on a lead. And the farmers at Dumyat will shoot dogs that worry sheep (and there are notices to tell you this). You can go over towards Aberfoyle, walking across the Menteith hills, there is a ROW over to Callander or you can just walk the ridge of the Menteith hills, it is pretty rough in places. Over at Glen Finglas the Woodland Trust have opened up new walks at the base of Ben Ledi, and I hope to go and have a look at them soon. Aberfoyle itself has a number of walks, the Doon Fairy Hill is short but interesting. You can walk for miles in the forests around there, but it is monoculture and a bit boring. You can go over to the WHW and walk sections of it.
Callander, there are the Craigs, Bracklinn Falls, up into the hills behind Callander, Ben Ledi, Ben Gullpen (one of my favouite walks), and around to Invertrossachs.
All this and you still have Perthshire, the waterfall walk at Comrie, Lady Mary's Walk at Crieff and the hills.
I have a working border collie by the way.