Short answer: Do the centre first and the Quay second, because Exeter is built on a hill and you want to be walking down it rather than up at five o'clock. Book the Underground Passages before you travel and check they are not closed — they shut Monday to Wednesday.
Book the Underground Passages. Tours take fifteen people and they close Monday to Wednesday, which is the single thing most likely to derail a day here. If your day is a Tuesday, build the plan without them.
Start at the cathedral green, which is free and open and gives you the building from outside. If you are going in, there is an admission charge. A free Red Coat guided tour is the better-value hour and you just turn up — several themed routes run and they cover the city properly.
The Underground Passages are a few minutes from the cathedral. Allow an hour. The optional crawling section is the part children talk about afterwards.
Istanbul Restaurant is open 11am to 11pm daily and the free breads and dips arrive quickly, which matters if you are working to a timetable. Babylon is the highest rated in the city with generous portions, open midday to 11pm.
Walk down to the Quay. This is the right direction of travel — the climb back up from the river at the end of a day is the mistake people make. Take the hand-pulled ferry across the Exe for 40p, then walk along Riverside Valley Park or out along the canal towpath as far as the time allows.
The Prospect Inn is on the quayside and very dog friendly, closed Mondays. The Custom House Visitor Centre is free and indoors if the weather has turned. If you have energy left, the canal path to the Double Locks is flat the whole way.
Exeter is the Devon day out least affected by weather. The Guildhall indoor family garden is free and dry, the passages are underground and unaffected, and the centre has covered shopping. A wet day here costs no more than a dry one, which is not true of most of this county.
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