I live near to Bath. There is a good train service, as noted, in terms of time to arrive, but you really need to book ahead or travel at the weekend to avoid being totally fleeced by the fares. For example, just pitching up in London Paddington this Monday and asking for a single ticket to Bath Spa, wanting to just get on any train that day would cost you £79, standard class. Presumably much the same back the other way.
And yes, I think you would tire of Bath after two days. It’s a shame about the lack of car because some of the other cool places in that vicinity like Avebury and the Cotswolds are just not served by trains. There are buses to Avebury from Swindon (not far away, same train line as the London-Bath line), though. I recommend Avebury. Especially if you like little villages surrounded by prehistoric massive stones. If you like those, it can’t fail!
Alternatively, you can catch a bus from Bath and go to Wells. Wells is very pretty but possibly lacking in sustained excitement for more than a long afternoon. Although if you’re a fan of the movie Hot Fuzz, that was filmed all around Wells so it has that going on.