I also support recommendations for east coast line with visits to York and Durham. Also, IMO Berwick-upon-Tweed is a lovely coastal town which could also be taken in with a small detour. You could break off journey in Edinburgh for few hours to take in a walk up the Royal Mile and the tattoo on the castle esplanade.

Without knowing your interests it is difficult to be absolute in recommendations so a word also about west coast line options. The scenery on this line does not become interesting until north of Preston/Lancaster. Should you go this way you could get out at Preston for football museum as already suggested and/or a side trip to Blackpool. Blackpool is perhaps the most famous of the old Victorian seaside resorts in UK and it may or may not be to your tastes (check google). Lancaster has a castle and Williamson Park with butterfly garden and a ''Taj Mahal replica'' at top of hill. A little further north you could also consider a stop at Oxenholme to do a quick jaunt around lake district. The scenery from Lancaster through to Carlisle is lovely. If you are romatically inclined and lucky to get availability you could stop in Gretna Green just inside the Scottish Borders to retake your marriage vows in the stable ;-) But I'm not sure this will be possible on a Saturday. If you are not worried about time then I recommend you depart the express train at Carlisle and take the Scotrail local service to Glasgow via Dumfries. The Nith River Valley is beautiful to pass through on train (sit on left side facing direction of travel). If you want to do a dinner stop then both Carlisle and Dumfries are pleasant small towns.

In conclusion, east coast line is the more obvious choice, however, depending on your interests then west coast line also has attractive options.

Have a good trip.