You can get the map through sustrans and there are more than enough sites with gpx files on to give you an idea.
I live in Newcastle so I rode to the station got the train over to Whitehaven then road to Tynemouth before heading home.
I know people who have used the transport people and for a group it was cheaper than the train and you don’t have to play roulette hoping that other people aren’t also wanting to get on.
http://www.chain-events.co.uk/new_c2c.htm
This guy might be able to help (Based in the East) as might Saddle Skeddadle.
From memory it is about 120miles on the road. There are plenty of opportunities to stop at cafes if you are doing it in a day.