We've covered a lot of that area at least four times.
I'd recommend traveling between mid-June and mid-August or be prepared for the possibility of snow.
Cooler temps in the higher elevation is normal so take your leathers and be prepared for cooler temps.

 

As far as the ride over the bear tooth goes i would leave Cody in time to eat a early lunch in cook city (the garlic burgers are really good) then ride over the bear tooth into red lodge. Better chance at finding someplace to stay for the night that won't break the bank

 

From Sturgis, I would recommend stopping by Devils Tower. Then, once you get down to Gillette, take hwy 14/16 - 14w instead of I90 (nice ride) .to Sheridan Wy. From Sheridan, take 14 to alt 14 (A14) over the Bighorn range towards Cody . From Cody take Chief Joseph (296) to Beartooth, Beartooth east to Redlodge, Beartooth is good going both directions so you will enjoy the ride back over the pass to Cooke City, then Silvergate, then into the NW entrance (Tower Jct) Yellowstone NP. Yellowstone is HUGE you can easily spend more than day looking around. ( Old Faithfull geyser, Yellowstone INN, Yellowstone Falls, Buffalo herds, etc. )
If you have the time, the Grand Tetons mtns are just south of Yellowstone, a trip thru the Teton NP using the park road is recommended. Just south of the Tetons is Jackson Wy. (some call it Jackson Hole) which is another place that's intesting if you have never seen it before, (Million Dollar Cowboy Bar, the square with the elk antler's, etc).
I hope this helps, that area is great riding.

 

This right here.

The bighorn range is beautiful riding with almost NO ONE around. The range doesn't last long but it's a great break from the monotony of the grasslands before you hit the eastern edge of the Yellowstone caldera area.

Stay the night in Cody then head up Chief Joseph and double track on Beartooth before heading into the park. Then I would suggest enjoying the wildlife herds, checking out Tower Falls, and then riding to Mammoth Hot Springs and checking that out. If you're tired, look for lodging just outside of the north entrance. If you're not that tired you can push on to West Yellowstone where the lodging options are more plentiful. Then you can hit up the stretch from West entrance to Old Faithful. This is where the majority of the park traffic is and the majority of the "sights" to see. Grand Prismatic Spring is a must-see, so is Excelsior Geyser...even if it hsn't erupted in like 80 years. You can also easily spend 2-5 hours at the Old Faithful geyser basin...2 miles of numerous active geysers, both small and large with 3 or 4 that they can predict. Grand Geyser is awesome if you can catch it...but it was only going off once every 12-14 hours when I was there this past august. Old Faithful and one of the others are under 2 hours between eruptions. Many of the small ones will just be going off as you walk by.

East Yellowstone area around fishing bridge and the lake is pretty, but gets congested with a lot of traffic due to those foreigners stopping to take pictures of the grasshoppers...and every single buffalo they see...and there are over 1000 in the park... If you're short on time, I'd skip that area and just head south towards Jackson and the Tetons over the Rockefeller Memorial parkway. But if you do have time, the ride to and from the park to Cody was probably my favorite riding in the Yellowstone area...better than Beartooth imo...but my opinion is from the perspective of someone who rides in Colorado year round so i'm a bit spoiled when it comes to high mountain passes .

Like someone else said, the best riding is generally outside of the park. The park itself is mostly the flat lands inside the super caldera and all of the cool, nerdy but beautiful geothermal activity that goes on there.

 

My wife and I ride Yellowstone at least once a year. This year we rode there in late July and October. We like to stay in Cody and enter the park through the East entrance. It's a nice twisty road once you get in the park. Past Fishing Bridge we take a right and follow the Yellowstone river towards Canyon Village. We continue north towards Roosevelt Lodge then take the Lamar Valley road to the Northeast entrance. Ride the Beartooth to Red Lodge and get some lunch, then ride the Beartooth back up and return to Cody on the Sunlight Basin aka Chief Joseph highway.
If you can find lodging in the park it will afford you more time to explore the area. The little cabins at Canyon and Lake areas are affordable and we stayed once in the motel at Grant Village.
If you stay in Cody visit Pat O'Hara Brewing Company for dinner, you won't be disappointed.
Have a great trip
Bill

 

From the Rocky Mountains west, you almost can't pick a bad road for riding in this part of the country. Just keep in mind that civilization can be sparse in places like Wyoming and Montana and keep an eye on your fuel gauge.

If you can swing it, time-wise...go south from Yellowstone and check out southern Utah. The scenery will amaze you, you will not regret it.

 

You may as well cross into Canada seein as you'll be so close...nice mountains to see and your cash will be worth 30 cents more on the US dollar.

 

I would also mention that a really good pair of binoculars is good to have in Yellowstone. This will allow you to possibly observe some wolves. They need to be a nice pair because the wolves don't exactly like people.

 

Coming down off of Bear Tooth which is a fun ride, even in the ivy rain like it was when I did it after Sturgis this year, you end up in Cooke City, great little town, Stayed at the Cooke City Apline 2x nice rooms and reasonable and just up the street is a great little restaruant, cant remember the name but it has outdoor seating and big flowerpots. Food has always been really good.

If you have the time I would highly recommend shooting up to Missoula and taking the 12 over to Lewiston. One of the most beautiful stretches of road I have ridden and lots of great twisties. It is 130 miles along the river. Here is a timelapse video of my run.

 

We did that exact route last year but we came up out of the flaming gorge in Utah from the south to Jackson and so on. Guess it's the exact route in reverse. I would have to agree that it is an excellent route. The Virginian lodge in Jackson,Wyoming was a cheap great place to stay and was very motorcycle friendly for us. Nice little bar and grill next door to with great food and cold beverages.