More general info
Cambridgeshire's navigable river system is the River Great Ouse. The end of this connects to the Middle Level (partly navigable fen drains) which connects to the River Nene at Peterborough. 37 locks upstream the Nene connects to the canal network in Northampton via a flight of 18 locks up to the Grand Union. Transiting the Nene takes about a week (quicker is possible, but it's hard work and places to stop are very pretty!), the Middle Level is usually crossed in a day and a half, so if you wanted to go and visit canals, it'd be 10ish days each way... maybe if you want a longer holiday? Also lots of marinas on the Nene and on canals in Northamptonshire which might still work for being nearish friends and family, but the public transport won't go to Cambridge.
The river itself is really and has three broad section:
The Bedford Ouse runs from Bedford through St Neots, Huntington and St Ives through pretty meadows and small towns and a few slow electronically operated locks. The towns all have a good selection of pubs too. This is the nicest bit for actually boating and has a lot of marinas.
The Ely Ouse is a big, wide lock-free stretch through empty fenland, which connects Cambridge's River Cam to the sea. Ely is great, Littleport is fine, Cambridge is full, everywhere else feels like the middle of nowhere. There's a reason why Upware's pub is called Five Miles from Anywhere! A couple of tributaries out towards Norfolk have pleasant, very rural moorings you need a car to access too
The Old West River north of Cambridge connects the two. Really more of a tiny fenland drainage channel that feels a bit like a canal and passes only a couple of tiny villages, some trees and the occasional cow. There are quite a few narrow boats here, but it's further from Cambridge then it looks on the map, especially if you're hoping for a bus.
Public transport from Huntingdon, St Ives and St Neots into Cambridge should be good, and Ely and Littleport has a rail link. I think Buckden Marina would need a connecting bus, and have no idea what the buses are from places like Upware or Pike and Eel or Fish and Duck Marina but suspect the answer is "not good" or "none". Based on what you've said about wanting public transport access to Cambridge and a pub, I'd probably go for Hartford or Huntington Boathaven or one of the ones in Ely or littleport. Rivermill Marina in Saint Neots is a cracking location too but no idea how to get in touch with them and doubt they have space (other St Neots marinas unlikely to have space for a liveaboard narrowboat either)
P.S. It sounds like you're quite early in your search, but just in case you happen to be actively looking to buy a narrowboat, I'm in the process of moving my narrowboat out of Cambridgeshire to sell it - let me know if you'd be interested in a look before I do that!