Which do you want - cheapest or fastest, it's rare you can have both.
Also when do you intend travelling as that may affect the price and which route is the best.
But have a play with nationalrail.co.uk
Cheapest means using the fare type known as "advance fare". Note that this type of ticket has zero flexibility, so if you want cheapest you have to be very sure of your itinerary.
Any tickets you book in advance, you need separate tickets for Bath-York and York-Edinburgh if you want to break your journey in York.
There's no material difference in time or cost between a direct train to York, and a journey involving changes in London or Bristol: the direct route is exceptionally slow, while the lines from London to Bath and York are among the country's fastest. But there's a HUGE difference between prices for travelling at different times of day, and between the price of tickets booked as far ahead as possible.
Also note that it's not an "either book an unrefundable inflexible ticket weeks ahead or pay through the nose on the day" choice. Prices do generally firm the closer you get to the day of departure, but some cheap Advance tickets may be bought up to about 1800 the night before - though these tickets can sell out at peak times of the year, like at Christmas.
You need to use the National Rail site to look at ALL time options.
I'd go via London, for most of the route via Bristol you'd likely be on the cross country voyager services which are horrible uncomfortable smelly trains
Returning to London is neither necessary nor advisable. There are hourly trains from Bristol (a very short ride from Bath) to York. That's for sure the quickest and easiest route; the whole journey will take around four hours. What fare you pay depends on the time you travel and, as rc says, if you are willing to commit in advance. That train runs all the way from Plymouth to Glasgow and stops at most major towns, so it is not an express. Last time I used it, it was a modern diesel railcar system and quite comfortable, so none the worse for that.
From York, you are on the main East Coast line with hourly express trains to Edinburgh. Journey time around two hours. Although a full-price single might allow you to break your journey overnight at York, it makes no sense to do that as you may well be travelling at times when cheaper "off-peak" singles (a walk-up fare) are valid, and again, if you commit to a specific train in advance you will be able to make very substantial savings.
Thank you for all the helpful replies! Planning to travel on Dec 21st or possibly the 22nd. I have looked at the national rail website but I am new to train travel so would love any suggestions or guidance!Edited: 27 September 2012, 15:58
You're a few days too early to book tickets for 21st and 22nd December. Wait a couple of days and they will become available.
The 21st is a Friday and cheap advance tickets might be a problem to get hold of.
I just checked on the Friday before you want to travel and the cheapest advance ticket from Bath to York is a staggering £101.70.
However, the trick is to buy a cheap advance from Bristol Temple Meads to York. The cheapest available on 14th December from Bristol TM to York is £46.00 compared to £101.70 from Bath Spa to York.
To get from Bath to Bristol just buy your ticket on the day of travel which will cost £6.60 off peak (after 9.30am).
So that will cost £52.60 instead of £101.70.
As has already been suggested have a look onwww.nationalrail.co.uk in a couple of days time when the tickets for 21 and 22 Dec are available.
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