I have reserved two HF Holidays trips, the first ending at 10 a.m. on Sept. 1 at Bourton-on-the-Water and the second beginning at 4 p.m. on Sept. 2 at Freshwater on the Isle of Wight. We expect to take the ferry from Lymington Pier to the Isle of Wight's Yarmouth. I'd like to see Bath. Does it make sense to go from Bourton-on-the-Water to Bath on Sept. 1, overnight in Bath and get to Lymington Pier at midday on Sept. 2, then catching the ferry to Yarmouth? We will travel by bus, train and taxi. If Bath is not a good idea, how about Reading for the overnight?
Your trip is perfectly viable but you need to go the short distance to Moreton-in-Marsh to catch the trains. (MIM is also served by trains from London). To go from MIM to Bath, you would probably change at Didcot.
When you go from Bath to Lymington Pier, you probably change at Southampton & Brockenhurst. However, when doing this journey, you may wish to break the journey at Salisbury and wander down to see the medieval cathedral. (Salisbury also has shuttle buses going the short distance to Stonehenge).
With the trains, you could buy one way tickets on the day BUT sometimes these are cheaper if you split the ticket at a calling point en-route - even though you stay on the same train throughout. This also applies if you buy Advance tickets for specific trains (non refundable). These can be very cheap and are usually released around 10 weeks out.
Play around with this website to see what it tells you:> https://www.traintickets.com/?/
However, there is an alternative which you may like to consider. Take a Pullhams Bus (801) from Bouton-on-the-Water to Cheltenham . Unfortunately, the 801 does not go to Cheltenham Railway station - which is a good mile from the town centre. I think a local bus does that mile or so. Then you could go by train to Bath OR go by train to Cardiff for a day or 2 before going to Bath or even day trip to Cardiff or Weymouth for example from Bath. I say this because there is a Rover ticket for sale that covers these lines at reasonable cost. It would also allow you to hop off on the Bath to Lymington leg at Salisbury. See > http://www.railrover.org/pages/freedom-of-severn-and-solent-rover.html
https://www.pulhamscoaches.com/bus/
If you did purchase the Rover ticket & went from MIM, you would have to pay for a separate ticket from MIM to Swindon - see Rover Ticket route map.
Pulhams also operate a service form B o W to Kingham service 802 which connects with trains to Reading from there
see www.pulhamscoaches.com