Wife and I (before we were married) did a three week trip by car around these areas. Start & finish at San Francisco.
Yellowstone certainly worth spending some time (several days) in. Look for the crowds at the side of the road as an indication of wildlife, etc. (Moose, Elk, Bison. We never saw any bears, but Yogi and his friends are there!) Old Faithful Geyser, lots of sulphur/mud ponds, fumaroles, to see, waterfalls, etc. If you want to back pack in back country, you will need a permit and take precautions against the bears, etc.
Bryce Canyon well worth the visit. We actually did an early morning helicopter trip through it when the sun was low and that highlighted the rock formations. I thought it was expensive at the time ($60/head but that was in 1987). Also lots of deer about if you drive to the canyon very early in the morning, before the hoards of tourists arrive.
Death Valley & Furnace Canyon were interesting. It was a holiday and we nearly ran out of petrol (or should I say gas!) Living on our nerves as we wouldn't want to spend the midday at the salt lake (lowest point in America) in furnace like temperatures.
Yosemite is very worth a visit and take a walk up to Twin Falls. Spectacular scenery and if you had time, a walk up Half Dome would be on my agenda (there were guide ropes, but don't do it if a thunder storm is pending, as lightning can travel down the guide ropes I understand
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Zion NP. Take a walk up the Narrows. Very impressive and the water is not too cold. Lots of signs there, that they may be inconsiderate to release a load of water upstream without notice and lead to a sudden rise in canyon water levels.
Grand Canyon (south side) was impressive, but we didn't have time for a walk to the bottom (3000 ft) or even a mule ride down. Very busy also and we had difficulty finding accommodation anywhere nearby.
Snake River Canyon if on route.
Sequoia NP as mentioned already.
Hoover Dam worth a look.
Maybe a night in Reno or Las Vegas. Food and accommodation were cheap, as they make their money from the gamblers. Maybe also take in a show there. Lots of old names appearing there regularly.
Grand Teton NP to drive through for superb scenery.
We never book accommodation and usually just drive until we find a cheap motel. Accommodation and food was certainly cheaper than that typically in Europe and the UK.
I would have liked a bit more time. Five weeks sounds pretty good. I would book accommodation for the first and last nights only. It's a PITA if you cannot find something close to the airport and have to drive miles to the next urban centre, followed by an early morning flight back.
A German friend did a two week holiday two years ago, to the States with a group of Germans (maybe 20 or so) and they hired Harley's over there and had a guide. That cost about €5k in total, but the guide and the Harley hire would be big ticket items.
Sounds like a good trip. Enjoy.
Grey Beard
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In fact, the Four Corners is all pretty amazing - follow the Green River.
