Born and grew up in Exeter. There's a couple of rough roads up around St Thomas, and i'd try to avoid Burnthouse Lane and Rifford Road but bar that, it's a really lovely city.
You might find property/renting cheaper if you go just outside of Exeter, places like Tiverton/Crediton/Honiton are still within easy driving distance but you aren't paying for the convenience of living in a city.
Gym wise, there's Fitness First but i'm not sure what the pricing is like. My brother raved about the one near the Iron Bridge in Exeter though, that might be worth checking out as he's quite fussy. Studio or something I believe, it's quite tucked away.
[Edit] This one:
http://www.yelp.co.uk/biz/studio-one-to-one-fitness-exeter