We went last summer. We were going with American family, so the flights were for that reason.
Heathrow to Washington (with a 3 day stop off in Iceland with IcelandAir which added nothing to the cost)
Day to recover.
Washington to Bozeman (change planes in Denver)
Bozeman gets you about 1.5hrs from the west gate at West Yellowstone.
We actually stayed further south in Idaho (beautiful AirBnB) but a bit far from the park.
Recommendations:
You need to more than a week.
Accommodation in the park books up far in advance, book now.
Accommodation outside the park also books up, but you then have to add travelling time. It takes about 20mins to drive into the park from the West gate. It's a huge park.
Go see the classics, Grand Prismatic Spring, Old Faithful, Yellowstone Falls, Lamar Valley (good for wildlife). All of these can be seen easily by car.
If you are up for a hike, you can spend days hiking to some of the more remote places.
Take a private wildlife safari - this was a gift and rather pricey, but great to have a guide take us in at 5am for sunrise and he spotted wildlife miles away that we wouldn't have seen otherwise. But that said, we still didn't see a bear, in fact didn't see one all week.
On that matter, you can hire bear spray, essential if you are hiking.
Classic tour bus routes - these go from the Old Faithful Inn, we did one at sunset which was pretty good.
Lastly, if you have time, drive down to the Grand Tetons.
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Grand Prismatic Spring - quite hard to get an angle to show it at it's best from the ground, but there is a higher view point that is supposed to be better. Use a polariser to cut down reflections and get the colour.
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Lower Falls from Artists Viewpoint.
Gear wise, I'd just take a 24-105 and a 100-400 or longer. Even at 400 some of the wildlife was too far away.
Plus either a good pair of binoculars or a spotting scope.
Take very comfortable footwear.
We missed loads of the park, didn't have time to do much hiking even though we were prepared for it. Driving between places in the park takes quite a while, the distances are big, and there's another amazing view just around the corner, so you keep stopping for more shots.
Enjoy the trip.