Short answer: The woodland below the moor is the sheltered option and it has four car parks. Parke, the National Trust estate on the edge of town, is the easiest walk from a car park in this area.
Ancient woodland in the East Dartmoor National Nature Reserve, with grassy glades, a river and a hide, and four car parks to choose from. Trees take a lot of weather out of a walk and the going underfoot is far better than open moor on 1 January.
The National Trust estate on the edge of town — the easiest walk from a car park here, free for members, and it works in any weather.
Haytor and the open moor are a short drive up the road, if the weather is kind and you want the view.
Almost everywhere opens later than usual on New Year's Day, and some places do not open at all. Ring in the morning rather than assuming.
Becky Falls and Canonteign Falls close over the Christmas and New Year period, and several Dartmoor attractions reduce hours sharply from January. Ring rather than assume.
Dartmoor on 1 January: the light goes by four, the weather turns faster than it looks from the car park, and the roads are unfenced with ponies and sheep on them. Start early, take a map, and tell someone where you are going.
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