Short answer: The canal towpath runs flat and traffic-free south out of the city, and the quay and Riverside Valley Park are free. Exeter is the easiest city in Devon to walk off a New Year in.
Flat, traffic-free, running south out of the city. You set your own distance and turn round whenever the party has had enough, and there is no mud to speak of.
Converted riverside warehouses on both banks. Short, pleasant, and the better end of Exeter to walk in.
Meadows and river paths running south from the quay — more open than the towpath if you want space.
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.
The cathedral, the Underground Passages and the museums are all likely closed on 1 January. Exeter's advantage is that it is weatherproof the rest of the year; plan this day around the water.
Exeter is the rail hub for the county, but New Year's Day services are heavily reduced. Check before relying on the train.
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