There are moorings available at Bridgwater Docks, though there is no dry dock on the canal, nowhere agreed with BW for boats to be craned out for blacking, and frequently not enough depth for cruising at the Bridgwater end without getting stuck. There's a great community of boaters though, and it could be a great canal if BW were able to invest more in it.
The Tiverton canal does not allow residential moorings, bonkers really, though it saves the council the costs of water points, elsan disposal, licence enforcement, dry dock/cranage points etc, and perhaps they consider the revenue from mooring and licence fees insufficient to cover all that.
The Somerset Levels are cross-crossed with huge 'drains' and canalised rivers that look great for navigation, 2m deep and upto 60m wide, but there are too many weirs, sluices and low bridges. Shame as the area was once rife with boating, as several 'disused' canals on the OS map and rotted wharfs attest to.
The Exeter Ship Canal, according to the harbour master, did not allow residential moorings either, though one of the liveaboards there told us it has been the watery equivalent of a common for centuries, with anyone able to put a boat in. "Just pop your boat in, there's nothing they can do about it", he said! There didn't seem to be anywhere near enough electrical points to go round though. That was 2 or 3 years ago though, and the council were planning to 'redevelop' the Quay and Basin (the liveaboards were all in the basin). No doubt residential moorings were not a part of their plans (again, the cost/benefit analysis problem, or the common developer mindset that canals should be made appealing to everyone except boaters).
The RYA instructor I did my level 2 with a couple of years ago said there were liveaboards in Plymouth Sound, though how 'official' that was he didn't know. Plus you'd really want a Dutch barge for that environment. There are at least two navigable rivers to explore though.
The nearest 'normal' place for a narrowboat to Exeter is the Kennet and Avon canal, though it's a long commute! Its connected to the rest of the system and is a popular place for hire-boating and liveaboards which means its a more obvious target for BW's investment (= proper pumpout facilities, independent boatyards, water points, marinas etc).
Hope you find something that works for you.