Since you have been to London, then there are some place that you might want to explore further in Scotland. First, you could go to Glasgow. This is a dynamic city with interesting museums, good food, good shopping, and a strong museum scene. You could also do some interesting day trips from Edinburgh. Stirling is wonderful and you can get to Bannockburn by public transit. I drove to Dumfermline and to Culross, but you can get to Dumfermline by train and I am sure that there is a bus to Culross. Both are very interesting historically.
The other Scottish suggestions would work better if you rented a car. One would be to visit the Trossachs. There are some lovely glens to explore. You can rent a bike, take it on the steamship down Loch Katrine and then cycle back. Rob Roy MacGregor's supposed grave is in this area at Balquhidder. The Lake of Menteith is beautiful early in the day and you take a boat ride over to see the priory. Another area to explore would be the Borders. You can see the Abby's, Sir Walter Scott's house, Traquair house, and Floors Castle. The Debatable lands have some very interesting tower houses and Hermitage Castle.
Of course, you can just take the train south and stop at York. It's a fascinating place to spend a few days.
Pam