I think you already know what you want to do.
Anton, you are answering your own questions. I think in your mind you already know what you want to do, and how to do it.
To minimise the chance of missing attractions, scenic routes and places of historic significance, etc. get hold of some good maps and/or a road atlas. Pick up a State issued map at the Welcome centres when you enter a new State. Most states issue them free. While there, pick up some brochures which interest you. Look for attractions along various routes on your maps, research a route or place on the web and see what their tourism people say. Always visit a visitor bureau or rangers office, or any other office where you know they have information about an area. These folk can often tell you about obscure places which are not necessarily on the web or on a map. Use all the resources you can think of, and keep a log of what you have learned, where you want to go, etc. Mark them in order of priority, so you do not spend all day at one place and have to miss out on a more interesting place further up the road. For this reason it is also worthwhile to have several different maps. I travel with a Rand McNally road atlas, AAA maps and pick up State issued maps wherever I go. Occasionally you will find something on one, which is not on the other two. My gps is used for finding addresses, and can't always get them right. I would not trust it on a multi State route.
Buy food at grocery stores, get a small campstove, so that when on the road and you want to cook up something - even for lunch - you can. Most rest areas have picnic tables. I always carry an old table cloth to cover the table, they are not always clean. Get the basics, knife fork spoon a small saucepan and a small frypan. I bought all those in shops like Goodwill. Even now when I find I want something, I go to Goodwill, and usually find it. Lots of ways to save money, check out the special forum for it.
To keep perishables, either get yourself a small fridge or get a cooler for which you will need to buy ice most days. With a fridge you would need an extra battery in the car., so that the fridge can run overnight without flatting the starter battery. I use mine to run my computer as well. Depending on your budget and the time you think you will be enjoying it, I would highly recommend the fridge, as you do not need to worry with all that water all the time. But they come at a price. However, I use mine at home even when I am not camping. Just going to the grocers in summer, perishables go into the fridge till I get home. That way the ice cream does not melt, nor the milk go off if you happen to be caught in a long hold up. You might like to explore the various brands. Mine is a Dometic. But you may be lucky enough to pick up one second hand. There is a Youtube video of the Dometic/Waeco 35 (which is the one I have), but unfortunately it is not in English. However you might find that a 25 or even an 18 would be big enough.
If you are looking at buying a new car, look for one that you can sleep in if need be. Some SUVs have fold down seats to give a flat floor. I know the Dodge Grand Caravan does, and have seen quite a few folk use that as a camper.
Lifey