Exmouth is the terminus of what is known as the Avocet Line which follows the Exe Estuary for its final 11, or so, miles, until reaching the sea. Trains run from Exeter St Davids and Central with a frequency of 2 an hour from about 6 am until 11 pm and with a journey time of just under half-an-hour.
This is quite a pleasant journey as the trainline joins the river soon after leaving the Exeter conurbation, passing through the villages of Exton and Lympstone. I'm not sure how much the fare is but my great value,10 Pound Devon Day Ranger covered me for the journey to and from Barnstaple (including a pleasant beer stop at Topsham!).
The railway station is on the edge of the town centre, less than 5 minutes walk, and about a 10 minute walk to the seafront. It is next door to the bus station from which there are regular buses to destinations around the area.
Ist website is for trains, 2nd for buses.
For a day out from Exmouth there's a regular ferry service across the estuary to Starcross, which is a place I haven't as yet been but is due a visit. From Starcross you can then get the bus or train onwards to Dawlish, Newton Abbot or Plymouth. This is a foot passenger only ferry (although bicycles are welcome for an extra pound) and the trip across the estuary takes about 15 to 20 minutes.
Timetables depend on tides and weather and the service runs from Easter to October - see website for details.