I was just there last September, it’s lovely. It sounds like you are planning on staying at hotels at night, then driving out to places to hike around during the day. Is that correct?
Climate will be hotter, clearer, and much drier than the UK. Bring a camera, big hat, sunscreen, and quality boots. I prefer lightweight long sleeves and long pants to ward off the sun and sharp rocks and plants, but that’s a personal choice. Culturally, most folks in Utah are quite friendly and yes, you can get beer easily.
The good news is that you can walk as little or as much as you like. There is so much to see regardless of whether you stay entirely on pavement, on trails, or go off trail entirely. Good maps or guidebooks will give detailed descriptions of elevation changes along trails.of course, Rangers will be the best source of information. As long as you are in average health and drink lots of water, you shouldn’t have any trouble beyond what you’d normally experience from lots of walking.
Try to be somewhere with a good vista at sunset whenever you are able.
Monument Valley is mostly a drive-through, but nice to see. Arches is easy to get around and spectacular in places, but may be crowded that time of year (not that it should stop you). Bryce is also gorgeous, but bear in mind it is quite a ways from the Four Corners area.
In addition to the places you’ve mentioned, I think that Canyonlands, Hovenweep, Natural Bridges, and Mesa Verde are very much worth seeing.
Canyonlands is huge and if you are not planning on camping out and/or renting a serious off-road vehicle, the Island in the Sky district of the park is your best bet. Easily accessible, not crowded, and some very rewarding walks. Upheaval dome, Grandview point, and Aztec butte would be worth a visit even if they weren’t surrounded by the rest of the park. You can get a great view of the Needles district from the overlook at the end of Needles Overlook Road (County Road 133, about 32 miles south of Moab on State Route 191). Another option to see Canyonlands is rafting, which will require an outfitter and is not likely to be cheap, but can be fun.
Hovenweep is a little out of the way, but it’s fairly small, so you can combine a visit there with another site (like Natural Bridges) or stop while passing through. The big attraction is the ruins, which I love, but I understand are not everyone’s cup of tea.
Natural Bridges is also not huge, but it’s just beautiful. You can see the whole thing by car in 2 hours or spend the entire day hiking into it and along the canyon floor. Also, the drive out to the park is very scenic.
Mesa Verde is THE place to see ancestral Puebloan ruins, specifically, stone structures built under overhanging cliffs. It’s big, and greener than the eastern Utah parks. It may be too far out of the way since are planning to do Bryce and Arches, but you do have 2 weeks. Bear in mind that part of the park closes around Labor Day, so if you do go, check to see what’s open.
Moab, Utah is very close to or within a reasonable, scenic drive of Arches Canyonlands, Hovenweep, and Natural Bridges. The Red Stone is a nice, clean, fairly cheap hotel a short walk from downtown.