Please, please, please don't even think of cycling on our roads - there are a zillion cars out there, these days, and a fair percentage of those are driven by mad people. Most our major cities are connected by motorways (freeways) and it's illegal to cycle on those (you would probably die before the cops picked you up, anyway!).
You might like to look at long distance buses (we call them 'coaches')- these used to be quite good value, although I haven't been on one for a while. For most coaches and trains it is usually cheaper to book sometime in advance (weeks in advance, that is).
Although Londoners moan all the time their public transport is much better than most of the rest of the country - they ought to live in Manchester, where I live, then they really would have something to moan about!
If all of this doesn't put you off, this is still a beautiful country and very varied for its size. Scotland and Wales are magnificent (as are their fine capitals, Edinburgh and Cardiff). My favourite parts of England are East Anglia, the West Country and the North. There is still a fair amount of folk music in the North, particularly in Lancashire and Yorkshire. These are both great counties with great people.
Generally, you should be prepared for a culture shock - lots of moaning, a very peculiar sense of humour and crap service in many places. If you have any romantic ideas about Britain divest yourself of them before you get here.
Oh yes, bring an umbrella - the weather is diabolical!