I used to live in Exmouth - it was lovely but just a bit big for us (we've since moved to village on edge of Dartmoor!).
There is loads going for families though - events, 2 x new soft plays (one in Ocean on seafront meant to be really good!), the beach obviously (I LOVED living near sea) and lots of parks etc. You can cycle along the Estuary too, which is brilliant. Bicton isn't far - soft play, tractor shed, train ride, big park, cafe, beautiful walks and picnic spots, very cheap for season pass - I really miss having this just down the road. Plus Dart's Farm, the park by The Puffing Billy. You've got the train into Exeter too - and boat trips in the summer. It was much easier to find entertainment in the winter months there than it is where I am now.
However, if you work in Exeter, the commute isn't great at rush hour! You can go the back way via Woodbury, or there's the train I guess.
There are some lovely villages around it - like Lympstone and Woodbury. They are pricey but for good reason! We heard great things about Woodbury school in particular. Lympstone is a beautiful little village and I know lots of lovely mums there. These spots are just that bit closer to Exeter and may give you more options with schools.
My friends have sent their children to schools all over Exmouth and seem happy with them. The Beacon springs to mind. I'm afraid I don't know much about them as we moved before my eldest has started.
We wanted to find a smaller village school and community, which is why we moved. Also to get more garden for our money! But we go back to Exmouth lots and I miss seeing the sea every day.
There are some lovely villages around Exeter with great schools. You can't really go wrong around here!!