With Dartmoor there's a couple of options, there are other thread about this but in short
Train to Ivybridge and within 10 minutes you can be on the Two moors way
Train to Exeter, bus to Okehampton and walk out (about 2.5 mile) to Meldon Quarry/reservoir which takes you straight on to the North Moor, or the Tarka trail path. Just make sure there is no firing on the ranges
You might also be able to get the train to Newton Abbott and a bus to Bovey Tracey but maybe Princetown
I am sure some of the 'Proper' locals will give you other ideas.