If you go to London, you want either a contactless debit/credit card or a pre-paid oystercard.
This will cap your daily payment to £6.50, no matter how many tubes/buses you use.
https://tfl.gov.uk/fares-and-payments/contactless/what-are-contactless-payment-cards#outside
Travel by Tube, it's much less hassle than buses and taxis are pointless.
National Museums are free in the UK, so go to the Natural History and Science museums. The National Gallery, Victoria and Albert, and Tate Modern are good for art and also free to enter.
Tipping is for taxis and 'proper' restaurant meals where you are served at your table by a waiter and get the bill at the end. About 10% is normal.
We don't tip for bar service, though a lot of coffee shops will have a tips jar. They are generally only used to get rid of inconvenient loose change. Most people will pocket the change and not tip at all.
Most cities are as safe as USA equivalents. Be sensible and you'll be fine. Don't leave bags unattended and keep your wallet away from pickpockets.
Outside of London, all the provincial cities have stuff to do - but most don't really advertise for tourism, so do a bit of research unless you know some locals.
Edinburgh is an exception to this rule and is basically London with different accents and tartan/bagpipes replacing redcoats/stupid hats. It's a lot smaller than London and very friendly in my experience (definitely worth a visit).
Glasgow has an unfair reputation for being rough and uncultured. It's actually a very nice city, but it's not as touristy as Edinburgh.
Train journeys can be expensive, so coaches (cheaper) or hire cars (faster) might be better, depending on where you're going.
I'd recommend York. Other than being a great historical walled city, it's also quite central, so it's easy to fit into a travel plan (especially if it includes London and Scotland). York has serious ancient history, so it's something that just doesn't exist in the USA. It's a very small city and easy to find your way around.
Cornwall is worth a visit in the summer, but transport is shit. There are lots of small pretty coastal towns/villages. Hire a car if you're feeling brave and can handle incredibly narrow roads (and be very careful with Cornish sat-nav). It might be hard to fit a trip to the south-west into an itinerary, but maybe you could loop through London-Bristol-Cornwall-Cardiff-Liverpool-Lake District-Scotland or something similar. You can stop off at Stonehenge on the way to Cornwall.
If you want to get away from cities, the Lake District is good for country walks and has plenty of cheap hostels.
Don't forget that Dublin is only a quick/cheap flight (or possibly ferry) away. Likewise, you could go to Paris for a couple of days if you're feeling more adventurous.