Town is very beautiful, great situated, but actually there is not too much to do in the night, and appart summer. But even summer is not so great. Beach is often impossibile cause of tipical English weather, but still i had there nice time.
What is worth of seeing is Dartmoor, two hours by car from Exmouth. If u are in Devon u should not miss it.
As well u can go to Exeter, nice city, they have free guided tour. Very interesting.
Night life in Exmount do not exist. Like in all british towns, every club is beilg closed at 1 am, more less and pubs even earlier.
But it is nice to stop there for 2, 3 days.