Given this is proper English countryside, Dartmoor is best visited if you have a car. Although being in the south west of England, in Devon, it is a long-ish drive from London, the Dartmoor National Park is well-served by a number of major roads.
M3 and M4 motorways take you south-west from London, where you can get closer following M5, which will lead you to Exeter - only about 5-10 miles from the National Park's borders. Other major roads are the A30 and A38 dual-carriageways.
The total driving time from London to the park (not to your actual destination) is ca. 4 hours (excluding stops) from the further Eastern side.
Much of Dartmoor is a true wilderness and the best way to experience it is to get off the beaten paths and walk - but bear in mind that walks should be well-planned and appropriate clothing and provisions taken with you.
There's nothing that you could really call a "main road" into, or across, the moor but rather a couple of major minor roads and a criss-cross of single-track routes. Unfortunately most visitors drive which whilst it makes the national park accessible does have the deleterious effect of clogging up the roads, especially during the summer, and rendering it difficult for the locals to go about their daily routines.
There are however over 40 bus routes into and around Dartmoor which include regular year-round services such as the 118 service from Okehampton to Tavistock, the 359 from Exeter to Moretonhampstead and the 510 from Exeter to Okehampton. In addition to these there are many less regular local services plus the seasonal services such as the #82 Transmoor Link which runs on Sundays, as well as some Saturdays and Thursdays, between Exeter and Plymouth
Many of these services are subsidised by Devon County Council, and some also by The Dartmoor National Park Authority, which keeps fares down and all have space for the less able-bodied.
Especially useful is the fact that most of the village bus stops are at the pubs!
First website has all the useful links. The second is an excellent interactive map of all Devon bus routes.