We travelled from Glasgow to Milngavie by the city metro - locally known as the 'Clockwork Orange'
Doubtful. The 'clockwork orange' (no one in Glasgow calls it that - it's an invention of guidebook writers) refers to the Glasgow subway which is a single circular line which only goes as far west as Partick.
A self guided trek through Scotland will be difficult by train. It needs careful planning, or a car.
It's pretty clear OP is doing the WHW, which is much easier without a car - even people who have cars with them have to backtrack on the train!
You want the trains out of the low level station at either central or queen street, which will also pass through a number of the local stations. If you are starting from Glasgow that day, there is no point in buying in advance. If you are coming directly to Milngavie from further afield, just buy your ticket through to Milngavie as it will save hassle.
Certainly book the trains ahead of time. You've already been referred to national rail. You may find that 'split ticketing' is cheaper - an example would be checking for the Fort William - Edinburgh leg whether it would be better to buy a single from Fort William to Glasgow, then another from Glasgow to Edinburgh (making sure that the second leg doesn't leave before the first!) rather than buying a single from Fort William to Edinburgh.
Your schedule looks... pretty hectic. Bath, for instance, is a four hour round trip from Fulham. Brighton's not much better. In Edinburgh, you will have virtually no time to look around - the early train from FW doesn't get in until 1pm, and if you get the second train, you will not be in Edinburgh till 5pm. The train to London takes 4 hours, so unless you get the sleeper, you won't have much time left for sight seeing in Edinburgh if you are only going to stay for one night. You appear to have left little time for sight seeing in London.
I think people on this site are sometimes a bit negative about train travel - you're not going to manage any of these journeys any quicker any other way (with the possible exception of Edinburgh to Paris), and the trains involve a lot less stress in my recent experience! Not least the fact that there are no limits on your liquids!