Well...
If your coming in at Heathrow:
I would get the Undeground into London which is cheapest but slowest, Heathrow Connect the middle one or Heathrow Express the fast but most expensive.
This gets you to Central London. Here I recommend Covent Garden: Street performers and the London Transport Museum. Looking to stay the night? Get the northern line to Edgware where there is Edgware Premier Inn which is very cheap and very nice and clean (weekends cheapest due to business location.) Others can advise fare ideas!
Other attractions in London: Natural history museum, British Library, London Zoo, Big Wheel and many many more depending on time in London.
Next, I would suggest an East Midlands Trains high speed service from St.Pancras station which is worth a look in itself. Here alight at Loughborough.10 mins walk through an estate gets you to GCR at Loughborough. A driving experience would be to Leicester and back to Loughborough.
Fantastic countryside? Take another train from Loughborough to Derby. (Another good Premier Inn - cheap and clean - here.) There is a local line from Derby-Matlock which is also East Midlands Trains. At Matlock, there is another heritage railway here to Rowsley and Darley Dale called Peak Rail. At Matlock or Rowsley, get a bus called Transpeak. This runs hourly and type it into google you will find timetable and fare details.
This bus gets you to Buxton. Amazing walks around here and a lovely town. Next, hotels in Buxton are available or a train to Manchester Picailly. This is Northern Rail and runs hourly. You could ride the light rail system, Manchester Metrolink here or a Premier Inn hotel stop again.
Here take a train from Manchester to Crewe then change at Crewe for a train to Bangor. This works out at £29 single. (Use national rail website.)
At Bangor, buses 85/86 take you to Llanberis. (Search for timetables on google.) At Llanberis is the FANTASTIC SNOWDON MOUNTAIN RAILWAY. This takes you up over 1000m to the summit of Mount Snowdon. If your into hiking, you could always walk up or train down or visa versa.
While at Llanberis, there are many good quality hotels around, namely a pizza hotel which I will found out the name for you. There is also a big visitor centre, tours of a mine or something and another preserved railway: Llanberis Lake Railway.
Then the 85 or 86 bus back to Bangor. Same £29 train back to Picadilly where you can take a Virgin Trains pendilino service to Glasgow.
http://www.topgearlive.co.uk/ You may of course miss this. (The studios are Guilford, though you have to apply to the BBC.)
If no Top Gear, high speed to London Euston via Virgin Trains. Then a quick underground trip to Paddington. Here you have the opportunity to board a sleeper service. One runs every night except Saturday and leaves Padidngton at 23:15 and arrives in Penzance, Corwnall the next morning. Supplements apply but it is an amazing experience. Penzance, Cornwall: You could take the 1/1A First Cornwall bus to Lands End (most SW point of the country.) Again good walking area. Or a FGW train to Lisekard where you can ride the picturesque Looe Valley Line to the fab town of Looe where Polperro, a car free old town is a bus ride away.
Aternatively, take the Caledonian Sleeper to Inverness, Scotland overnight which is also an austounding experience waking up to the Scottish Highlands and Mountains. At Inverness, take the Far North Line through amazingly scenic countryside to Wick/Thurso. Thurso brings opportunity to a ferry trip to Stornoway. Or Kyle of Lochas line to the west is just as good.
A high speed east coast journey to Peteborough from Edinburgh is a scenic route running right next to the sea. At Peteborough, a premier inn is avaialable and an EMT local service will take you to Ely with its amazing cathedral and shops. Here, an opportunity arises to explore the wonderful county of Norfolk by train, visting towns such as Cromer or Mundesley.
If your in Cornwall, Bomin and Wenford steam railway is brilliant, (accessible from Bodmin station.) Newquay branch nice also although in peak season can get very crowded.
There are many, many more rail highlights in Britain, including The Island Line on the Isle of Wight (ferry) which uses old underground stock and 2 steam railways on the breathtaking Isle of Man (ferry) and a load more.
Right now by hands are aching but hope this helps, welcome to the forum, welcome to our railways and have a good trip!
I enjoyed doing this so don't worry if you don't use it, but please take as many ideas as possible as I believe I have put the best highlights in!
EDIT: possible All line rover best? Premier Inn, Travelodge, Holiday Inn Express hotels are all motly good and can be found in every city. Travelodge can vary, so check reviews on Trip Avisor first.
EDIT: Coming into Gatwick or Stansted: Gatwick Express and Stansted express aso get you into central lonon.
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