Yes, your UK friends are now waking up!
If you walked into Liverpool (Lime Street) station today and purchased a ticket to Stratford upon Avon, it would cost you £41.30 for the 2 hour 37 minute trip with 1 change. (Click details at www.nationalrail.co.uk to tell you the change points. These are often only a few minutes - this is normal as most trains run to the minute). If you advance purchase this route, the price stays the same. But - we don’t want to pay all that! Looking at purchasing a ticket for tomorrow - we can get from Liverpool to BHM = Birmingham (New Street) for £9.50. (If you bought on the day of travel, it jumps to £29). Birmingham New Street is right in the centre of the city. It would be worth having a walk around for an hour or two. Then, purchase on the day a ticket from Birmingham (Moor Street or Snow Hill station) to Stratford-upon-Avon = £7.70. Total cost Liv to Stratford-upon-Avon = £17.20.
If you do not wish to visit Birmingham, you should check the times for a journey from LIV to SAV and click ‘details’. This will tell you to change on the outskirts of Birmingham at Smethwick Galton Bridge (SGB). Note the times and rapid change times for the train to Stratford-upon-Avon. Should you choose this option - still purchase (in Liverpool) a day or two in advance a ticket for LIV > Birmingham (rather than SGB). Also purchase SGB > SAV but make sure you tell the clerk the date of travel first.
If you walked into Liverpool station today to purchase a ticket for a train to Bath (today), it would cost you £82.70 for the trip - which is around 4 hours. I just checked for 11 October and the prices are £26.80 to £33.70. So, in this instance, you should definitely pre-book a specific train ahead.
No rail line directly links Stratford-upon-Avon with Bath. Therefore, taking a circuitous route, the trip would take around 3 hours with 3 changes and the price is £59. I would not advise doing this. Assuming that you have gone from Liverpool to Stratford-upon-Avon and then wish to proceed (on another day) to Bath; I would take local bus services via some quaint Cotswold villages. For example, today you could leave Stratford-upon-Avon on Bus 24 for Broadway, Worcestershire. (This bus goes via the lovely small town of Chipping Campden). For timetable > choose route 26 & click MORE.http://www.johnsonscoaches.co.uk/timetables/
Then, you could depart Broadway on bus 606 for Cheltenham Spa, Gloucestershire. Have a look around Cheltenham before taking a local bus for the 1 mile link to the rail station. The Broadway to Cheltenham bus service is here:> http://www.marchants-coaches.com/606-Service
Now, as it would be difficult to perhaps figure out an Advance train to book to Bath, I would just show up and pay on the day. This is likely to cost £14.30 and you just get on the first train that is going to Bath. If a Cross Country express arrives that is heading SW - get on that and travel as far as Bristol Temple Meads - where you can change for another train for the short hop to Bath. The journey from Cheltenham to Bath should take about 1 hour with trains every 30 minutes between Cheltenham and Bristol TM and every 15 minutes from there to Bath. If you pre-booked the Cheltenham to Bath trip, it would cost as low as around £10. It simply is not worth it and having the stress of having to get to Cheltenham station for a specific train.
Pay directly the local bus drivers. If you do not have pre-purchased rail tickets, buy tickets at station before boarding trains - unless boarding at an unmanned station.
Bath to London (Paddington) is cheaper if you pre-book about 11 weeks out.