Short answer: Exeter has an unusual amount of genuinely free things to do: the Red Coat guided tours, the Custom House Visitor Centre, the Guildhall indoor family garden and Belmont Sensory Garden all cost nothing. The cathedral green is free; going inside the cathedral is not.
Exeter's Red Coat Guided Tours are completely free, run on several themed routes, and you simply turn up. There is no catch and no expectation. For a city with as much history as this one, an hour with someone who knows it is the single most valuable free thing on offer, and most visitors never find out it exists.
Down at the Quay, free to enter, with a film on the city's history. It is also the start point for tours, so it works as the first stop of the day.
Free, indoors, and you can bring your own food in — which makes it the most useful wet-day stop in Exeter if you have children. Free nappies at the baby change.
The hand-pulled ferry across the Exe is not free but it is 40p, which is close enough. It is pulled across on a cable by hand and runs seasonally in daylight hours.
Exeter Cathedral charges admission to go inside. The green around it is free and open, and evensong at 5.30pm is a way to hear the building rather than tour it. The Underground Passages are ticketed and need booking ahead — tours take fifteen people and they close Monday to Wednesday.
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