Woodbury Common is fanstatic; about 3m N. of Exmouth. I grew up 10 minutes from from it. You’ll need a guide to help you make the most of it but there are a couple of great local groups. Without a guide you’re bound to find some great rides albeit with a few dead ends. The best thing about it is that getting lost isn’t a problem. If you pop out into a village there’s always a pub / cafe.
Dartmoor of course is excellent and spectacular and massive but don’t even try to attempt it when ‘off’ season. It’s just cold and miserable. It proved to me that sometimes there is the wrong weather not just the wrong clothing.
Good coastal footpaths all along the South Coast if that’s you thing. The East Devon Way and others. My parents completed the entire length of it (in stages) last year.
Lots and lots of cheeky trails E. of Exmouth around Sidmouth, Beer, Seaton and whilst cheeky, I never once had any issue with riding them.
Haldon – meh. Overpriced cafe, expensive car parking and not a single route I couldn’t ride along with a bike trailer. Fun little pump track though.
The Quantocks are good but have only ridden there a handful of times.
I’ve never made it to Exmoor.
I’d imagine that there’s some good road riding although it’s not really my thing.
I’d allow 40 minutes to Exeter (depending of course where you want to go) but it will probably take 30. The train only takes 20 and runs fairly frequently. If it’s a regualr commute for you, I’d definitely look at non-car options. Parking in Exeter is a right pain in the arse.
I have friends living in the Honiton area. It’s lovely. You certainly get more house for your money than if you were to move further West; the villages in Exeter’s commuting belt are some of the most expensive outside of London.