If you have time, an overnight train from Victoria Station in London is ideal. Must have cost me about 40 quid thereabouts and I had a first class compartment and woke up to breakfast and newspapers - the train ride was actually one of my fondest memories
I got around by taxis a lot - which must have been expensive. All the shops were down Union road, so you do not need to go to several places to do your shopping
We crossed from The Netherlands to Hull with the North Sea Ferries and took our own car. We drove up through Scotland to catch the ferry to Shetland and then on to Orkney.
Having a car is easy as you can see so much more. I think trains are expensive in the UK and buses are not always frequent in rural areas and also limit your travelling losing you a lot of time.
NAtional Express operate a directs ervice from London (and other UK cities and towns) to Aberdeen. Takes about 12.5 hours.
return price about £38.00
I found a promotional return for £20.00
GNEr operate direct trains from London (Kings Cross) to Aberdeen.
Fare £103.00, but you might get it cheaper if you book in advance and change a few times.
This photo was taken while on the train to Aberdeen from London. I'm not sure where it was other than we had already passed the Firth of Fourth, so we were between Edinburgh and Aberdeen.
Here is a crane of some sort that was obviously used for train switching at some point in time. I have no idea if it is still in use, but I just liked the look of something from another time....