I live in an extremely remote part of Scotland and travel often to London, via Edinburgh or Glasgow.
Both those cities are really easy and surprisingly quick to reach from down south.
East Coast trains go between London and Edinburgh.
Virgin Trains go between London and Glasgow.
ScotRail runs modern, clean, fast, quick shuttle services between Edinburgh and Glasgow (and vice versa), every 15 mins during working hours and every 30 minutes at less busy times. Journey time around 45 mins .
Don't be put off by seasick refs to Virgin Pendolinos. I use them often - and, while they may affect a few people, I am often sick travelling by road but have never been troubled by them. Am not a fan of what Richard Branson stands for politically etc, but really do I take my hat off to him re trains between London and Glasgow. The East Coast trains are good, but less luxurious, in my opinion. But for both you have to be canny:
a) book in advance and if possible online to get cheapest fares
b) Virgintrains website virgintrains.co.uk - will give you probably the best fares for west coast AND east coast routes.
c) expect sleeper trains to be very, very busy in July/August. People will have been booking them for weeks already. But if you can get a berth, your DD will probably love the adventure. It IS romantic - tho' to be honest it's quicker and more efficient to travel by day.
d) expect all trains, everywhere in the UK, to be busy on Friday and Sunday afternoons
e) RESERVE SEATS It's free and makes all the difference in the world. You can ask for reservations in the quiet coach, if that's what you like.
f) If travelling on Sat or Sun, consider Weekend First - only £15 extra, free snacks and drinks, and much, much more space and comfort
g) Get a FAMILY RAILCARD before you start to book - gives you big discounts
I don't live in Dundee but it has some very good things to see - eg Shackleton's Discovery ship and exhibition about the Antarctic, historic wooden warship, wonderful science exhibitions, great art gallery (and soon, an outpost of the V&A), old castles, country parks, heritage trails and sites...www.tripadvisor.co.uk/Attractions-g186518-Activities-Dundee_Scotland.html
Agree that Edinburgh Airpost bus is excellent - sets down and picks up right outside airport departures/arrivals and goes to city centre. Frequent, too. Edinburgh is officially one of the most beautiful cities in the world, with stunning views, historic sites, posh shops, lovely gardens, great museums and art galleries and - at festival time - the most astonishing range of entertainment, from staid or hip to really wierd and wonderful. At festival and fringe times (basically, August) there is a real buzz in the air, even if you are not actually going to any of the concerts or shows.