I usually stay in hostels, so I can’t say much about B&B’s, but as for things to do…
In London: My favourite tourist attraction is Westminster Abbey. The place has history seeping out of every stone, and everything there has a story attached to it. Make sure you fork out the extra money for a guided tour - it’s the best £3 I’ve spent in London. You’ll get so much more enjoyment out of the visit and learn about ten times as much.
The Tower of London is also great, although it’s a bit pricey (at least £10 from memory). But the towers are magnificent (the White Tower is beautiful in the sunlight) and the Beefeaters are very entertaining tour guides. And you get to zip past the Crown Jewels on a whizzy little conveyer belt.
The London Eye (an enormous wheel with glass pods large enough for about 20 people to mill around in) on the south bank of the Thames will give you a great view of London, and the trip around lasts about half an hour. I’ve heard it’s quite good at night.
Camden Markets are worth checking out, on every Saturday and Sunday. Colourful people, loads of really cool stuff, loads of junk, and a vibe so tangible you’ll have to have a bath afterwards. Catch the tube to Chalk Farm and walk south through the markets, ending up at Mornington Crescent tube (they close off Camden tube itself at the weekends.) If the weather is good (and at the moment it’s great!), grab some sandwiches and a couple of beers and sit in Regent’s Park or Primrose Hill, both behind Camden.
My favourite museums are the Science Museum and the Museum of London. The latter is like walking through a story, detailing the history of the city from ancient times to today. The British Museum has an unbelievable collection, but it’s not necessarily the most informative place. Pieces seem to be accompanied by a bare minimum of description that don’t really do much to put them into their historical context. But I’d still recommend going.
The Tate Modern is definitely worth visiting, and I loved the National Portrait Gallery (behind the also-worth-a-look National Gallery on Trafalgar Square).
When you’re in central London, make sure you walk everywhere. Don’t take the Tube. Everything is much closer than you’d expect. You can walk around Picadilly Circus, Liecester Square, Trafalgar Square, Whitehall, Parliament/Big Ben and wind up at Westminster Abbey in a couple of hours. (That’s a great introduction to London, by the way.)
Brighton is worth a visit - it’s about an hour and a half by train south of London. It’s got a great feel to it, nice beaches, it’s famous pier, and great pubs/bars/nightlife. Check out The Lanes on a weekend - dozens of back-streets full of funky shops, markets and stalls. Perfect for a day-tip or an overnight stay.
Everyone told me that I’d be disappointed with Stonehenge, but everyone was wrong. It’s beautiful and eerie and impressive and interesting, and everything else it should be. Just don’t expect for it to change your view of the universe and you’ll be fine. Another good candidate for a day-trip from London.
My biggest recommendation is to spend some time in Scotland. Edinburgh is possibly the most beautiful city I’ve ever seen. Spend some time walking along the main drag, Princes Street, with beautiful buildings and shops down one side, and gardens, museums, monuments and the odd Castle down the other. Climb Arthur’s Seat, an ancient volcano with magnificent views. (Don’t worry, it hardly every errupts.) Get drunk on a pub crawl along the Royal Mile, up to the Castle gates (no need to go in, it’s a bit pricey for what you get). Walk around the New Town.
Try to visit the Isle of Skye - I’d recommend renting a car in Glasgow (try http://www.easyrentacar.com) and driving up to Mallaig. The scenery is gorgeous and you will spend the entire trip with your jaw hanging open. (The other islands, such as Harris and Lewis, are also worth visiting, but I doubt you’d have enough time. Skye can be done in three days, including travelling there and back from Glasgow or Edinburgh.)
Don’t pay more than £6 for an English breakfast. And for £6 worth, you should be sweating lard and about to explode. If you have a Scottish breakfast, make sure your coffee cup is full before you try the Black Pudding. You’ll be needing a drink to wash the vileness away about three milliseconds after it touches your tongue and your body screams at you this it is Very Very Wrong.
And don’t forget to Mind the Gap.
Have fun!