This is very doable, but you may have trouble finding lodging. Start calling the parks' lodging reservation lines once a day and grab a room if you can get one!

Jackson airport is right between Jackson and Grand Teton NP, so it's not out of the way at all.

We spent a day and a half in Grand Teton and three days in Yellowstone in June 2010. Here's a link to my trip report:
http://www.fodors.com/community/united-states/grand-teton-and-yellowstone-june-2010-chilly-and-gorgeous.cfm

There are plenty of other western parks you could visit, but Yosemite will be slammed with tourists that weekend and the Utah/Arizona parks will be really hot.

Lee Ann

You have enough time for both parks. We like Grand Teton NP more than Yellowstone, but it would be a mistake to be a short drive from Yellowstone and not check it out.

Have you considered getting hold of a copy of one of the guidebooks for these parks--such as those published by Fodor and Frommer?

HTTY

Doable as everyone says--and I am like Happy Trails, but it is really close to take a tour in Yellowstone--and you wouldn't want to miss a taste. Problem of lodging as everyone says, so just get right on it.
You might check hotels.com.

You'll find that Yellowstone is much more crowded than the Tetons. Something to do with being more famous.

The Grand Teton Lodge Company (GTLC)is the official booking agency for inside the park. Accept no substitutes.
http://www.gtlc.com/

Jenny Lake is most expensive, followed by the GTL and then Coulter Bay.

I spent 6 months there as #2 person for all retail operations. Work hard play hard.

Grand Tetons are nice, Yellowstone is GREAT!!! You wouldn't catch me at either of them on the 4th of July weekend. Probably a nightmare. If you get a mile off the road then not so bad, but will be very very busy.

There are motels close enough to the parks that you don't have to stay inside them if you are not able to book park lodging.

The Grand Tetons are more beautiful, but Yellowstone is very interesting, unlike anyplace else.

Thanks for the great advice!

Call, Call, Call for inpark reservations.

People cancel all the time especially closer to travel date.

If you want to see wildlife other than bison they are most active early and late. You don't want to be driving too much in the dark. Never know what's on the road.

In my opinion, Yellowstone has the attractions such as geysers, wildlife, waterfalls, etc.

Grand Tetons are beautiful for the scenery and wildlife. Absolutely beautiful.

You can see photos of my trip to YNP & GTNP at:
www.travelwalks.com

Another option for lodging in GTNP - Signal Mountain Lodge. We stayed there and loved it. It has 2 restaurants on site, and a small grocery store. The views are to die for.

We spent 2 weeks last fall in YNP and GTNP adn enjoyed them both for different reasons. July will be very hot in YNP, and there is little shade there.

You can see my pictures here http://ioana-mark.smugmug.com/Travel/2011-Yellowatone/18866535_s7B33Q#!i=1463769139&k=xtjrQtd

Have a great trip! And call for YNP in park reservations!

We also stayed at Signal Mountain. Nice place and very central.

xyz99, good shots!!!

I am looking to plan a similar trip either for next year or 2014. I'm still trying to figure out what time of year to go as it would have to be during the summer. I would be very interested in where and what you choose to do.