Train

https://www.nationalrail.co.uk/

Go to that website. Fill in the required info. It will provide a list of trains on that date. Once you select a train, the website will send you to the appropriate site to book your tickets.

If you are unfamiliar with European trains and tickets, have a good look through the Man in seat 61 website. https://www.seat61.com/

Essentially two routes- Edinburgh to London Kings Cross, cross London, London Paddington to Bath, every 30 minutes
OR-
Edinburgh to Bristol direct, change, 15 minute train ride to Bath, hourly

The overall journey time is all but identical at just about 7 hours.I

Pick the right train and not much difference in fares either.

There are flights Edinburgh to Bristol on easyjet. Overall journey time about 5 to 5 1/2 hours.

When is your trip? It would help to know the approximate date.

Many tickets will have what are called Advance tickets, which can be much cheaper than the fares you will see on National Rail. I would look at tickets from London North Eastern Railway (LNER). https://www.lner.co.uk But it might be possible to split your tickets and get an even cheaper rate.

If you can post more information about your trip, there are British folks here who are very well versed in train travel planning and would be happy to help.

Thanks to all of you.

The usual cheapest split point if doing the west coast route is Stafford, not Wolverhampton. I have tried every possible combination before now. That applies for the South West or for London.
My own best, earlier this year was £29 railcard fare Paignton to the Cumbria coast (on one certain train a day from Paignton).
That was £9 Paignton to Bristol, £11 Bristol to Stafford (on quite a few daily trains) and £9 Stafford to home.
Only booked about 4 weeks ahead of travel and knocked the socks off all the split ticket websites.
That £9 southbound from Cumbria is usually found on the 0652 from Edinburgh. It is taking splitting to dangerous levels, but it is normally cheapest to split as Edinburgh to Carlisle and Carlisle to Stafford.
The cheap fares Stafford to London are normally on the Manchester to Bristol trains as opposed to the longer distance ones.
On the 0652 I like to sit in cars U or G on 11 car sets to avoid hordes of school children in Cumbria.