Secondly, from your planning map, you are missing Utah and Arizona,
Also , most of the interesting parts of New Mexico and Colorado.
I really don't like these road-trip apps. I know it feels like its giving a good start on planning, but its really giving you an itinerary that's so generic it borders on useless, and that you could have done much better on your own with just a little bit of research. In a sense, I think by putting together a product that looks and feels already complete, it subtly discourages users from digging in and doing better. And believe me, you can do better.
I have several questions right from the start:
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when is this trip? the seasons are a big deal, and will partially determine the outline of your trip. For example, there is literally no time of year where you can have a good visit at Big Bend in Texas, then drive immediately north to Yellowstone and Glacier. Yet, this is what the map has you doing. Some things like this will need to be adjusted based on reality.
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what type of experiences are you most interested in? Keen wilderness hikers? Mountain climbers and adventurers? Casual walking and sightseeing? Historical sights? Museums? Nightlife, cities, etc.?
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where have you already been in the US, and are you strictly seeking to avoid those areas?
What would be the most cost efficient way to travel?
Generally, I agree with above, a cheap rental sedan is the way to go for cost-efficiency. Big RVs are an expensive hassle. Are you interested in camping? In the western US a campervan rental won't save you much money, but it can make the camping experience more comfortable and convenient. In any case, camping or motels, we do a lot of self-catering on the road rather than constantly eating out. Pick up a cooler for perishables. We also enjoy this because it often forces you to find a nice spot to eat and hang out in whatever town you're in, like a city park. Blend in with the locals in their natural habitat.
Would it be a good idea to book all the accommodations in advance or is it possible to find cheap rooms a day before?
This depends on "when" and "where". Outside of the summer travel season, you can probably book as you go, except at the most popular destinations, where reservations are always a good idea. During the summer travel season, booking ahead most places is a good idea, and might be very necessary in some areas if you want to be at all picky about where you stay and how much it costs. There are few "last minute bargains", but lots and lots of "last minute hassles and unexpected expenses".