The shots you get will depend on the time of year and the time of day when you are there. Early morning is better for wildlife and mountain flowers. If you enter the park from Jackson, you will drive along the Snake R to the left of the road, and as you approach Teton, watch the left side of the road carefully for free ranging moose. They like to play in the Snake and browse the willows and other shrubs at river's edge. But they like EARLY. Much past 10:30-11 AM, they will be bedded down and more difficult to spot.
If you access from Cody, you will drive along the Shoshone R, on the left of the road. From 2-3 miles east of the Park entrance and into the park at Pahaska Tepee, the area along the river is moose and grizzly bear country. But you must watch sharply to see them.
Inside the park, Hayden Valley is wolf country, and the rangers will tell you if the wolves are active. You will see many buffalo, elk, some antelope, possibly a few mountain goats, coyotes, eagles and other raptors. But you must keep a sharp eye to see them. Take along a pair of good binos and have your passenger scan as you drive. Yellowstone Lake is picturesque and easy to get sweeping panoramic shots, and you'll often see small herds of buffalo and some singles near the water's edge.
As to gear, your 70-200 will be adequate but definitely not enough for the long shots. If you can, try to get your hands on something longer, like a 100-400, and a telextender. A 1.4 is good, but I usually end up using my 2.0. The longer you can get, the more good wildlife shots you will get.
If you are after mountain shots, there are those aplenty. Once again, to get the glacier on the middle Teton, you'll need your longest lens or lens/telextender combo. And, be sure to take along a solid and sturdy tripod for the long shots. If you find a likely spot for wolves, e.g., you'll set up your rig along the road and wait, possibly panning and looking. You'll need the same rig for coyotes, prairie dogs, ground squirrels, too. A hand held 100-400 will be ok for eagles, hawks and falcons on the wing if you have a reasonably steady hand.
I use my 24-105 and/or 24-70 for walking around, but when I go into the park, will always have the 70-200 mounted and sitting on the seat next to me or on my wife's lap - so I can grab it quickly.
Whatever you do, do NOT try to approach buffalo, moose or bull elk. They are dangerous if they get agitated, especially cow moose with calf. That's why you mount the 70-200.
Finally, the gate rangers are well informed on a daily basis regarding good wildlife viewing areas on the days you are there. Sometimes they post notices at the gate, but if not, ask them for that info.
If you are staying at Cody, try the Proud Cut Saloon, right on the main street, which is Sheridan Av, for good food and drinks in a real western atmosphere. In Jackson, look for the Sourdough restaurant, an old log house/restaurant a block off the square.
Have a good trip and enjoy. You will have tons of 'targets of opportunity' in that region.