Done this drive multiple times over the years. I'm from New York City. By the way, the vast majority of YNP is in Wyoming, with thin slivers of it in Montana and Idaho. It's a beautiful place, well worth the drive.
However, I really do hate to say it, but two days each way is a severe underestimation, especially if you want time to see things along the way. If you want to get there quickly, and even if you push EXTREMELY hard every day, allow at least three days. Even then, it's exhausting, perhaps to the point of danger. If you want to be there somewhat quickly without exhausting yourself to dangerous levels, allow four days each way, that's a safe estimate. To have time to stop along the way, allow at least five days, maybe six.
This is roughly a 2,400 mile drive. 35 hours might be correct, but driving 17 hours each day for two days is not a good idea, and it's dangerous. Keep in mind, allow at least 3 hours each day for food and rest stops. Therefore, to drive that distance in two days, you're talking about putting in a pair of 20-hour days. That's asking for danger. Don't do it. You will be driving drowsy (or even falling asleep at the wheel), which is often worse than driving drunk.
For a decent trip, allowing you to get there somewhat quickly while not risking falling asleep behind the wheel, figure on doing about 600 miles a day. In our years of traveling on family road trips, we've topped 800 miles VERY rarely in a single day, and that's even with LONG summer days with multiple drivers to switch off.
That said, I-90 should work all right for you, there's stuff to see along the way. It's probably among the fastest ways to go. You'll also pass through the Badlands/Black Hills region of South Dakota, worth spending some time there.
How long will you be able to be out on the road? Perhaps I can help figure things out a bit, give you some ideas of things to see along the way if you have time to do that. Add a detail to your question with some extra info if you want.
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EDIT: If you'd like, please open the option to receive e-mail through your profile, or use the link through my profile to do that. Just trying to find the best way to discuss this without typing too much on this page here. By the way, where are you starting from and where are you going to? Which campsite, for example? Could affect the suggested directions.
14 days, seven camping. So, Drive on days 1-4. Day 4 is your first camp night. Day 10 is your last. Drive back on 11-14. So, that's four days each way, which is manageable. Still, that's not allowing a huge amount of time to see much else, which is OK, because you get to really explore Yellowstone.
If you want to have time to see other things, maybe consider cutting the camping down to five days, allowing an extra day in each direction of travel (or two explore days in one direction, and straight driving in the other direction). Maybe explore the Black Hills a bit.
I completely understand not wanting to stop a lot, and if you get there in three days (even if you get there in four), you will NOT be stopping a lot.
If you're driving 17 hours, that's putting in at least a 20-hour day overall. Allow about 3-4 hours a day for food, rest stops, filling gas, and other delays (traffic). Those things add up very quickly.
As efficiently as my family has always moved on road trips, we find it difficult to cut the "added time" down to less than 3 hours a day for food, gas, restrooms, and delays. We average 600-700 miles a day when we push hard, but we'll pop an 800 once every few years. That's usually out west where you can do 70-75 on the Interstates.
I wouldn't recommend putting in much more than a 16-or-17-hour total day (driving and stops). That's borderline dangerous. I want you to be able to get there without worrying about whether or not you'll fall asleep at the wheel. As beautiful as the scenery is out that way, things can get deceptively monotonous, and you can just drop out sleeping without warning.
Considering a four-day drive (I can adjust this depending on your starting town and finishing campsite):
Day One: Finish west of Cleveland, but east of Chicago. This should be a good 700-mile drive, give or take a bit depending on where in VT you're coming from. Once you get too far west, you're getting into Gary/Hammond, Indiana, which aren't the greatest places in the world to be in if you're not familiar with them. After that, you're sitting in Chicago. So, stop before that. Maybe Toledo, Ohio...or Elkhart, Indiana if you're really flying along.
Day Two: Try for Sioux Falls, South Dakota. An early start is ESSENTIAL unless you want to take forever to get through Chicago. If you find you'll be hitting Chicago during the morning rush, avoid it completely. Maybe go via I-80, then cut up through Rockford. Heck, maybe just take I-80 all the way through Iowa, then cut back up to Sioux Falls from Omaha. This might be a good idea even if you're early or late for the morning rush; Chicago traffic is unpredictable, and can set you back two hours at any time of day.
Day Three: You might make Yellowstone if everything has gone PERFECTLY. Otherwise, you'll be within striking distance to get there on Day Four, with a leisurely half-day drive. If you're falling short, then maybe try Gillette or Buffalo, Wyoming. If you're falling shorter than that, maybe look at Rapid City, S.Dak.
Day Four: You'll be sleeping in Yellowstone, barring major problems such as car trouble.
Keep in mind, two of your days heading west will be 25-hour days, due to time changes. On the flip side, two of your days coming home will be 23-hour days.
As for places to stay, the major motel chains should work just fine (Holiday Inn, Best Western, Days Inn, Motel 6, and so forth). Many independent mom-and-pop places are good as well.
I recommend, if you have the option, not making reservations, but actually pulling into town when you're done for the day, and looking at potential stopping places to make sure you're stopping in a decent place. We've made the mistake of reserving in advance a couple of times, only to find we were in seedy parts of towns, where the motel desk had bulletproof glass, occasionally with dents. Not a good sign.
A membership with AAA is a great idea. Great for roadside assistance, which has saved us a few times. It's also very good because you can get their TourBooks, which list approved lodging establishments around the country, so you can get an idea of what's in each town. If you find you'll be stopping late, it might be a decent idea to call ahead to reserve a place a few hours in advance, especially if you're in places where larger towns are 20-30 miles apart. Furthermore, check the dates of Sturgis Bike Week. Finding a place to stay is nearly impossible within a 200-mile radius of Sturgis, S.Dak., while this rally is going on.
In short, don't bite off more than you can chew. You say the most you've done is a single 10-hour drive in one day; be careful not to stretch your limits to dangerous levels. I completely think you can do this trip, but you do have to be very careful about it. Two days is basically impossible while obeying speed limits and getting safe amounts of rest. Three days is borderline. Four is safe.
I'm being realistic here, that's all; and believe me, my idea of "realistic" is much more optimistic than many. I'm always the one in my family who says we can pull off certain driving tasks when the rest say it's tough. Heck, I talked them into driving to Alaska from New York. Still, this is one case where even I say it's stretching limits to do what you originally proposed here.
I'm a HUGE fan of road trips, and I'll scratch and claw for any mile to find a way for people to pull off a trip they want to pull off. I'm trying to find a way for you to get there in a safe amount of time, any way possible. I think this should work. Let me know if you need anything else, this should be a great trip, even though you'll be doing little else except driving on eight of your days. They do say getting there is half the fun, though...