You can cover Yellowstone in 3-4 days and could easily spend as much time in the Teton.
However, my best advice is to make reservations immediately! You may already be too late to get accommodations inside the two parks.

Reservations will be hard to come by. Look at everyone else's postings (do a search) about this. Many places were already booked up as of December.

You can keep calling the reservations office for cancellations though.

It will be our first time to Grand Tetons & Yellowstone this July as well! See you there.

Sorry - can't help with the mileage from SLC to parks. Try entering the driving directions on Mapquest or Mapblast for approximate mileage.

Jackson is right outside Grand Tetons NP (south end as you approach from SLC) - however, again you better get looking for reservations.

I fear that Howard R speaks the truth about reservations in Yellowstone.
The lodgings concessioner in Yellowstone is Xanterra. You might be able to find something at Grant Village for one or two nights.
http://www.xanterra.com/properties/yellowstone.htm

There also is some restricted availability of rooms at Mammoth.
If you want Old Faithful, your chances are not very good.
But move on it now.
In the Tetons, try Signal Mountain Lodge.
The lodge has a web site.

the drive from Salt Lake City to Yellowstone varies greatly depending on where you are going and how you go.
It is an all day drive regardless.
I don't recall the exact number of miles, but we drove it last summer via Logan Utah so we could see the Logan Canyon. It took us most of the day without any major stops to make the trip.
But, days are long in July in Yellowstone. Again, in Yellowstone you will take what you can find. The last time I checked around Old Faithful for myself, there was nothing available.
And trying to get two consecutive nights anywhere around the lake was not possible except in a very expensive room.

Bob Brown speaks the truth...plan a day's drive from SLC. Make sure you take the route over Teton pass. The view is breathtaking!

I'd spend a day or two in Grand teton first, then up to Yellowstone.

We stayed at Colter Bay Village...clean, rustic...and absolutely loved it.

Also, you need to go to Beaver Dicks's and the Mangy Moose at the Jackson Hole ski area, and a trip on the tram is beautiful!

Forgot to add that my son and his family stayed at Colter Bay village and liked it. If it has a vacancy, grab it. You might consider staying there and "commuting" to Yellowstone.

In my opinion, some of the cheaper cabins in Yellowstone are not very good. In fact, I would rate some of them one notch better than a leaky tent.

There is something about staying in the park that I like, particularly if you are close to Old Faithful.

If you can't get a place in the park (you will be lucky this late - last year we were lucky to get our choice when we booked in Feb), then consider staying some in West Yellowstone and in Gardiner (just north of the park). The park is so large that if you are continually driving from West Yellowstone, it will seem like you are doing nothing but driving.

If you can, I would forget about doing the trip this year and plan it for next year. You can make reservations at the park hotels/cabins up to a year in advance.

We stayed at the Old Faithful Lodge cabins and found them to be nice and very affordable

Thanks for all the wonderful replies!
Actually we just got 3 nights in Canyon Lodge & Cabins (Frontier Cabin). How's this place? Is it inside the park? (I guess it's right in the center of Yellowstone, right?)

Also, any good places in/close to SLC so we can visit as well?

Thanks again.