Short answer: The Prospect Inn on the quayside is the easiest with children — outdoor space, very dog friendly, and right by the water so there is something to watch. Samuel Jones has a big outdoor bar area that suits groups. Both are on the Quay, which is the part of Exeter that works best with a family.
Exeter's city-centre pubs are city-centre pubs: fine for an evening, less good with a four-year-old at half past five. The Quay is where the pubs have outside space, a view that occupies children for free, and room to stand up and move around without being in anyone's way.
Right on the quayside, very dog friendly, and positioned so that boats, the ferry and the water do the entertaining. It is closed on Mondays, which catches people out on a bank holiday weekend.
A big outdoor bar area, good for groups, with sport on the screens. That last part cuts both ways with children depending on the fixture — it is livelier than the Prospect and better suited to older kids than toddlers.
The Fat Pig does gluten-free Yorkshire puddings, which is a rarer thing than it should be, and it is worth booking for Sunday. Note it is closed Monday and Tuesday. It is a pub rather than a family venue, so it suits children who will sit at a table.
If the weather rules out sitting outside, the Guildhall indoor family garden lets you bring your own food and is free, which is a better wet-afternoon answer than squeezing a family into a busy bar.
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