Short answer: Exeter has three stations and the centre is entirely walkable. St Davids is the mainline one, Central drops you in the middle of the shops, and St Thomas is handy for the quay side of the river. Everything in this guide is reachable on foot from one of them.
Most Devon towns have one station or none. Exeter has three, and knowing which to use changes the day.
The mainline station, where the London and Bristol services call and where the Tarka Line to Barnstaple and the branch to Exmouth begin. It is about fifteen minutes' walk from the centre, mostly uphill on the way in and pleasantly downhill on the way back.
Smaller, and it puts you directly into the city centre by the shops and the museum. If you are coming for the cathedral or Princesshay rather than to change trains, this is the one.
Little used by visitors, but it sits on the far side of the river near the quay end of town, which makes it the shortcut if the Quay is your destination.
The Exeter Ship Canal towpath starts near the quay and runs south, level the whole way. Two of the pubs on it — the Double Locks and the Turf — are properly out in the fields, and the Turf cannot be reached by car at all. Walking or cycling there is not a compromise, it is the only way in, which makes it one of the few places where having no car is an advantage.
Exeter is the hub the rest of Devon hangs off. Exmouth is a branch line away and the station is near the seafront. Dawlish and Teignmouth are on the sea wall line south. Barnstaple is the end of the Tarka Line, which is the way into north Devon. Topsham is four stops down the Exmouth branch, or an hour's walk along the canal.
Dartmoor is the thing you cannot easily do. Buses run to the moor edge but services are sparse and seasonal, and the good walking starts where the buses stop. If the moor is the point of the trip, Exeter is the wrong base without a car.
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