I've traveled the length and breadth of England. I've been for a week plus more than four times. Some tips:
Book your trip about 2-3 months in advance. Skyscanner.com is great for finding the cheapest month/cheapest day in the month. I recommend Virgin Airlines. If Virgin is ~100-150 more, still book Virgin. Also, be sure to check prices flying in to Manchester or elsewhere, sometimes the savings can be significant. Sometimes a flight will have a layover in a random place, too, one time I flew in to Paris and then hit London, cause it was for some reason 400 dollars cheaper.
You can absolutely get away with no driving and I HIGHLY recommend it unless you want to go very far off the beaten path. I rented a car every time and every time I regretted it. On one of my trips I got a flat tire and it was a shitshow trying to get things straightened out. If you do rent a car, you must request an automatic. I drive stick, but trying to drive stick on some of the insanely tiny roads of England with a hand you're not used to shifting with is unfun. Also, be prepared to pay for parking EVERYWHERE and sometimes it's extortionate. The rules of the road are fine but sometimes you'll be expected to drive at some breakneck pace on lanes separated by a sparrow's dick. Then in a lot of towns there are assholes parked on the sides of the road that are designed for like 1 horsedrawn carriage, and you have to drive on the other side of the road to avoid hitting them, pissing off oncoming traffic. If you try to stay cautious and safe it just pisses the fuck out of everyone. It's just not as roomy as US driving and it can be stressful in certain places--freeway driving was fine. Anyway, I ubered everywhere in main cities that was too far to walk, and took the train and bus everywhere else after my car got a flat and it was extremely freeing. Do that.
A really good guide for places to avoid in cities can be airbnb. Get airbnb most places (though in the North, a lot of times hotels are cheaper and nicer). Read google reviews of hotels and user reviews of airbnb places. There will absolutely be families and parents that call a place out for being shady. I loved my airbnb in London and in a few places in the north got really charming, cheap hotels. Most tourist stuff will be totally fine. I've done seriously almost everything in Glasgow, London, Manchester, Liverpool, Cambridge, Leeds, Bristol, Brighton, and more, and never felt unsafe. I did accidentally one time walk into a very shady area in Leeds but that wasn't a tourist destination.
You can't legally carry anything. Don't bring a knife. Walking stick is fine. It sucks cause my everyday carry includes a knife and, when I'm not teaching, a sidearm.
I recommend the British Museum in London HIGHLY. It's very similar to the Met and absolutely stunning. If you're a museum person I'd dedicate a day and a half to this, and if you're not, at least 3ish hours. I cannot recommend the Winston Churchill museum. Overpriced and short. I liked Glasgow more than Edinburgh, I may be in the minority there. Glasgow has a pretty cool downton music scene on the right days. I had an absolute blast listening to live music and going from bar to bar. Liverpool has an entire area that is Beatles-themed if that's your bag: I did it mostly so I could say I did it. I found it overpriced but the statues of the Beatles and fair nearby were cooler than the expensive tour. The beaches on all sides blow cock. I spent a lot of my life in southern California, and the worst beach there is better than the best beach I went to in England. A lot of them are stuck in the 70s and have serious economically strange models--like even if you go at a certain time everything is closed, cause there's some 3 week window of a tourist season where a bunch of gambling and fish and chip shops are open. The cliffs of Dover are pretty awesome.
More general tips: be prepared to pay for things you wouldn't in the US. In my opinion their bathroom culture is fucking baffling. I had to pay for a bathroom several times. Public bathrooms close at odd times and are never open late. Sometimes restaurants don't have bathrooms. Almost nowhere, including Mcdonalds which I ate at a few times out of necessity, do not offer free refills which they are bizarrely proud of. I get thirsty when I eat and sometimes I want a refill of water or diet coke or whatever. Waiters don't expect tips which is cool I guess, but the service is generally something you would expect at a shitty Applebees. I drink mostly water and I had to ask to get my 5 ounce water refilled (with no ice) frequently. All of the places you'll eat will kind of expect you to get a pint with your meal. Oh, and nothing is open after 6 except restaurants and pubs, and the further north you go, the more it's just pubs. You'll see people get shitfaced with their meal and walk (or drive) home. Their drinking culture is very different from ours. A lot of places will have difficulty with your American AMEX or Visa or whatever, and some places will handle it just fine. Bring cash everywhere and be careful with your wallet on the tube because someone WILL be pressed against you if you travel anything resembling normal hours. Be sure to call your card companies or go to their websites and issue travel dates (where and for how long) so no charges are blocked. Also some cards charge for international purchases. Some charge a %, some a flat fee, some both. I had a card that was free on that and used it exclusively.
England can be a great place to visit and I do love it there, if I seem negative. I think my number one tip is: make a really good plan. You can have open times and how much of that depends on traveling style, but pre-book everything. For example for a 10 day trip, I'd book say 3 days in London with the London Pass to see a ton of attractions and an airbnb which I have researched and is a 5 minute walk to a tube station, then a train to Liverpool and a hotel there, I'd uber or lyft to the places I want downtown, take the tour, maybe a day there, then a train to Manchester or Leeds, maybe a few days there, then up to Edinburgh or Glasgow. When I was there last year there was a killer deal on a train from London to Glasgow for some great price, but all of it had to be done in advance and with planning. Same day it was like the same as a plane, planned a month off it was like 40 pounds. It all depends on when you're flying in, what kind of traveler you and your daughter are, and how much you're willing to spend. On a 1-2 week trip I'd limit yourself to London, few spots in the North, one to two cities in Scotland, and one to two places in Ireland. Anything more is going to be rushing it, and on a one week trip I'd cut at least one of the places. Take tours, find out what you like and if you prefer the Big City tourist stuff or the small city tourist stuff. I'd say the best trip to England/Europe you take is always the second trip after you know what you like, but you can have a great time the first time if you plan it right.
