Just to chime in about visiting Bath, my daughter and I spent a few days in England in August before heading to Belgium and then France. We took the train from London for a day trip and it was definitely worth the cost (which was not remotely close to $140 for round trip) and also the time. The Roman Baths are fascinating (even if one has been to Rome). And any fan of Jane Austen will love seeing the Pump Rooms and Royal Crescent, among other things.
However I will say that we'd seen enough after about 4 hours because it got VERY crowded in the afternoon (was a Saturday), plus it was unseasonably warm for England. (Although who are we to talk, since we're tourists and were part of those crowds.) Anyway we were really glad I'd purchased tickets that could be exchanged for a small sum to take an earlier train back to Paddington.
We had a similar experience in France when we took a day trip to a small town and it wasn't what we expected. On that occasion I had not purchased tickets that could be amended and so I had to basically buy new tickets so we could get back much sooner and make the most of the rest of the day in Paris. Lesson learned, always buy amendable tickets if possible. (Except in cases of traveling on the Eurostar, for example, when seats can apparently sell out in advance. And we loved the Eurostar -- it was our first time on it, going to Brussels via the Chunnel.)
And as far as the Tube v. the Metro in Paris, with the smartphone apps for each, they are both super easy to navigate. I much prefer the Oyster card though to Paris Metro tickets. Way easier to manage and you don't have the issue of tickets being demagnetized. That happened several times during a week in Paris.
(And just FWIW, on our previous trip to Europe a year earlier we took the train to Portsmouth for a day trip. Seeing the ships (especially the HMS Victory) made it totally worthwhile, and it's a cool little town.