Difficult one really, from the point of view that although there are lots of rail companies there's seldom much choice to be had, as they often don't directly compete.
Chiltern Railways has always had very good press, but unless you're interested in immersing yourself in deepest Buckinghamshire there's not all that much of a reason to travel with them.
First TransPennine Express; they have nice trains really, but the risk of you meeting myt332 is unacceptably high.
Where to start... People have always gone on about Settle to Carlisle, which I've never done. Newcastle-Berwick-Edinburgh is pretty impressive. Actually, no... start at York, as then you pass over Durham viaduct for one of England's classic railway views.
There's not much to see on the Circle Line on the London Underground...
It depends really what time of year you're thinking of being around. Some places are quiet out of high (summer) season but absolutely unbearable once the tourists start coming in. It also depends how long you'd like to stay in one such place and what you'd like to see. Some of the quaint little villages in the UK are so because they have no access to the railway; if you time it right local buses are usually the best onward route for a day trip to somewhere the train won't take you.
Unless you have a particular fondness for power stations, don't let Didcot's easy rail connections from London Paddington fool you into thinking there's much to see there. To be fair, it has a little railway museum, and one of the highest life expectancies in the UK. Just not much else, except the aforementioned power stations.
Alton Towers? Haven't a clue!
[Edited 2009-02-20 10:19:27]
Have you ever felt like you could float into the sky / like the laws of physics simply don't apply?