In nine days split between the two parks, you can hit the highlights, see amazing country everywhere you look and have an incredible vacation.
Nine days is enough to tour the "Grand Loop" in Yellowstone. I would spend four days exploring Yellowstone. Based out of est Yellowstone and staying a night or two in the park if you can get a reservation.
I don't know about lodging. I have an old camper van, and and use the campgrounds.
Though i has been a low snow winter, there will still be a lot of snow around and hikes are pretty confined to lower altitudes. The geyser basins around Old Faithful are a great place to wander.
Weather in May is a crapshoot. It is just unpredictable. Might be sunny and 70F or it may snow. Click on "more about jhscott". I have made several posts on weather that contain info links.
The key is to be flexible. Have options if the weather stinks. Local museums, the Grizzly Discovery Center in West Yellowstone. I think they have an IMAX theater too.
To see the Tetons stay in Jackson. Lots to do regardless or weather. It's off season and rates are as cheap as they get.
Come on out! Other than lodging, don't make any specific plans. Wing it, tou'll have a great time.
JHScott gave great advises on YNP and GTNP. For your visit to GCNP only GCNP south rim comes in mind because north rim still will be closed till may 15. So you have to drive from Jackson to GC Village (SR). Keep in mind this will be a 2 days drive (over 800 miles) one way. So if you plan to do this you better start flying to Jackson and depart from Phoenix or even better, reverse your route. Please let us know where you are flying to and from. There are many awesome sites like the Moab area, Bryce, Zion eo closer to YNP.
I missed the part about Grand Canyon.
If you try to do all three, you will have minimal exposure at each or the three, and spend most of your time driving.
The Grand Canyon is just too far away.
Haha JH, you are right but recently there was a family from Maryland who wanted to drive from MD to the Black Hills,SD and visit Badlands NP on their 2nd day after the drive. The drive was 1500 miles so I said, maybe make it a 3 days drive.... But is was no problem for them to drive 1000 miles first day and 500 miles second day and still would visit Badlands. You never can tell how far someone is driving in one day.
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