I might have already done a similar prompt, and lord knows I’ve waffled about trains before, but to answer the question: I’d choose the train for any form of cross-country trip.
Why? Because it’s the most relaxing and most scenic form of travel (well, usually). If you’re driving, you have to maintain a certain level of awareness and alertness, you can’t switch off from keeping your eyes on the road and surroundings. Sure, you can stop over here and there, but you can’t really enjoy a sight-seeing trip as a driver. Granted, I speak from a place of ignorance on that, but that’s how I would feel.
With trains, you can sit back, unwind, enjoy the company of your friends (or if travelling alone, read a book, listen to music, play a game etc), and take in gorgeous landscapes. Depending upon the service, you would get food and drink delivered to you, or you can pop to the buffet car. The railways usually link the big cities too, so you can jump on and off at different locations to take in different sights.
All of this is dependant upon a set of circumstances where you have a fair bit of money for such a venture!
Cycling has its benefits, though for a cross-country trip… well, even cycling across the UK from east to west would be a loooong trek, so be sure to pack for the occasion! I’d need me a lot of snacks…
Flying is the quickest way, but it comes down to the reason for the trip. If you need to be from A to B for an urgent meeting, you fly. If you want to enjoy a leisurely journey, take the train.