You certainly can find parking in NYC, though it’ll be quite spendy, I imagine. When we were in Los Angeles in May, my husband got a free hotel room (from frequent-traveller points) - but we had to pay 50 dollars a day for the car. Whatever hotel you plan to stay it, contact them and find out about the parking situation.
If driving looks likely, reserve a vehicle now. I don’t know what the rental car situation is right now, but a few months back it was pretty grim - people paying a fortune. When we took our big train trip this past May, I reserved it 6 months in advance. The guy at the rental agency said he had someone who reserved a car, cancelled it, then tried to rebook - and the price had doubled.
New England: Well, anywhere you’ll go, you’ll find pretty scenery. You’ll find a lot of older buildings etc, that you won’t find in other parts of the country, because, well, NE was one of the first parts of the country settled by Europeans. Mount Washington, in New Hampshire, is interesting - highest point on the east coast, and FREEZING COLD even in high summer. If you visit, take the cog railway, which is worth it just for the experience - you can drive, but frankly I’d be white-knuckling it the whole way. I’ve been twice: the first time, I was wearing a light jacket, and birkenstocks, and nearly froze; my hair was blowing around so much I had trouble taking photos of the kids. The second time, we were driving from Vermont to Maine, and stopped almost-on-the-way as my husband had never been there. It was rainy and foggy and basically no view at all.
If you were interested, you could piggyback a jaunt into Canada from a New England leg - there are quite a few border crossings into Quebec from northeastern NY thorough Maine, and from north central NY on west into Ontario.
Ocean: The Atlantic coast is pretty. Might be too cold to swim in the ocean that time of year - when we used to go to the beach in Maryland, the water wasn’t really comfy until late July / early August. Further south, it’s swimmable earlier, so if that’s an interest, you could visit the coast in South Carolina. Further north, it really isn’t EVER good for swimming - when we went to Acadia National Park in July, the water was something like 60 degrees. We dipped our feet in just to say we had, and that was all. The Pacific will be too cold for swimming, per my own experience, anywhere you’re likely to be - not sure how far south you’d need to go to find water warm enough to swim in without freezing off anything important, but it was too cold for us near Los Angeles.
Glacier National Park will be fine in the summer - maybe a bit crowded though. If you want to stay IN the park, make reservations as soon as possible as they tend to book up. Kalispell, MT is a good alternative - near the western end. When we visited, in 1986, we stayed in the park, but the last night we stayed in Kalispell as we had to drive to Seattle the next day. Similar advice if you want to visit Yellowstone, Grand Teton, or either side of the Grand Canyon. There is not much near the North Rim, accommodation-wise (I think there’s a lodge in the park and it might be easier to book a place there, as it’s much less crowded).
We did a trip in 2010, where we flew into Las Vegas, drove the next day to Kanab, Utah, and made day trips to several of the parks (Zion, Bryce Canyon, and the North Rim) as they were all just about 2 hours away. This was on the advice of some Dopers, when I posted about the trip back in 2009 or 2010. As noted, the more popular part of Zion REQUIRES a shuttle (if you’re staying in the park, that may not be true). If it’s a warm day, one of the outermost shuttle stops lets you off at an area where the river is quite shallow, and lots of people were splashing around in the water - I wished that we’d brought swimsuits! Water shoes would be nice for wading there, to protect your feet from the rocks.
Depending on your route: there are rafting trips that put in at the bottom of the Glen Canyon Dam, in Page, Arizona, that take about a half day: you float a few miles down the Colorado, then get bussed back to your starting point. We did that in 2010 and really enjoyed it.
As far as driving time per day: There will be a number of parts of the trip where there’s nothing of any real interest and you just want to go nonstop. Be careful of getting fatigued and drowsy - caffeine can only do so much. I’ve done 700 miles in a day solo, but do not enjoy it (and in my case, it requires pharmaceuticals). My husband and I did a few 700-mile days on our trip (back when the speed limit was 55 mph, so that was a minimum of 14 hours), and at least once on that trip, we had no damn business being on the road: both of us were dangerously drowsy during the last 100 mile stretch. We should have stopped sooner - or at least pulled over and stretched our legs - but we had in mind that we had to get to such-and-such town. We were damn fools and lucky to have come away with nothing but memories of that leg of the trip. Anyway - even on the “nothing of interest” days, consider limiting yourself to 6 to 8 driving hours if you can.
Especially in the western parts, make sure you have snacks and beverages in the car so you can pull over for refreshments. Plus, on the off chance the car breaks down, you’ll be very grateful to have drinking water. One of your first stops, once you hit the US, should be a store where you can purchase a good ice chest, and lay in a small case of bottled water (even if you generally use refillable bottles; this would be for emergencies). Getting stuff like sandwich fixings can be a huge cost savings - that was how we did breakfast and lunch on our driving trip. Towns are fairly sparse in places.
Anywhere west of the Mississippi, if your car’s gas tank is below half full, make sure to top it off at your next stop. Consider making a stop at that point even if your stomach is full, your bladder is empty, and you’re feeling quite alert. If it gets down to a quarter full, actively look for a place to refuel. Gas Buddy (smartphone app) is pretty good at finding gasoline (and shows prices). If you see a “last gas for xx miles” sign, check your fuel gauge carefully - I sailed past such a sign on our trip, blithely saying we had enough, without glancing at the gauge - then shortly after, saw we only had a quarter tank. We made it to our destination, but on fumes. That was another STOOPID moment on that trip.