I was toying with the idea of visiting the US either next year or 2026, and not only is the northeast the cheapest destination to fly to, but to my limited knowledge offers the most things to see. (unless you can persuade me in favor of, idk, the midwest, the south, whatever)
What I was thinking of visiting:
The Empire's HQ (D.C.) - obvious choice. Imperial architecture, containing probably one of the most famous museum complexes in the world.
Annapolis, Maryland - looks like a quaint seaside town (or maybe Baltimore, MD, with its antique ships in the harbor. Though I hear it's a bit rough)
Harper's Ferry, West Virginia - John Brown! Plus, it's in West Virginia - an excuse to cross one more border.
Philadelphia and Boston have old colonizer architecture and legacy stuff... but to my understanding, basically nothing is left. No idea if there's anything on indigenous culture in that part of the country.
New York City - One of the cities, for good or ill.
I'm definitely also going to New Haven, CT. I have no idea what's there, but I have something to prove to a comrade (or get owned) 
Also I'll probably have around a week, and I should have my license by then, so was thinking of renting a car. Seemingly, the car brain is strong enough that rental fees are surprisingly low.
Ya so as others have pointed out a week is not enough time to visit all these places. It would be like trying to see all of europe in a week. It would be a more enjoyable experience to focus on a specific area. NYC can be fun and has a lot to do but its very expensive. Expect to spend like 500-800 USD per day and thats with a cheap hotel and not eating at any fancy places. If your like dead set on it you could fit max like 2 or 3 cities into a week but your gonna be missing out on a lot, and spending hours traveling you could be having fun during.
If you really wanna do it tho this is what id suggest as someone who has seen a good bit of the east coast myself.
Fly in to the nothern most city you want to see, probably Boston it sounds like? Spend maybe 1 or 2 days there, take a BUS to NYC(you can arrange to have a couple hour layover in philly to get a cheesesteak if u want). Buses are THE budget travel option in the US. Greyhound, flexibus, etc. Way cheaper than a train and faster usually too. Spend a few days in NYC, then BUS to DC. Youll arrive at union station. Its massive and beautiful. Big marble work, and vaulted ceilings. Awesome architecture. The national mall and museums are the only thing really all that worth seeing in DC imo. You can hit up the ones you want to see in a day or so.
To burst your bubble a bit you mentioned getting your drivers license so i assume your young. In the US you have to be 21 to rent a car. So if you arent yet then thats a no go regardless of if you can drive. Uber and rideshare is probably gonna be your fallback in that case.
Id try to spend the most time in NYC as its the place with the most to see and do altho if your really into historical stuff maybe DC will be more your thing. Or Boston ive never been to Boston so idk how historical it is. You can find lists of attractions for each city online and choose ones you like.
Honestly tho you didnt mention it but if i was gonna see the US for the only time and had a week id go for the natural wonders of north america not the man made stuff since that u can find anywhere. But stuff like the grand canyon, yellowstone, the redwoods, etc are pretty unique.
800 Km is not that much, especially if you divide the route between several days, unless you travel on rural roads with low speed limits. But yeah, it's probably wise to shorten the amount of driving necessary, as a passenger or not.
Also, I'm gonna be 26 next autumn. I didn't get my license earlier because it was not really necessary and I didn't have my shit together anyway. Now I am employed and have a long commute, so driving would take around an hour and a half daily off it.
How feasible would it be to stay in say Newark and take public transportation to NYC? To my understanding, the surroundings of NYC are cheaper.
Very reasonable, assuming you take PATH
Very doable, I've done it before (but not recently, so can't comment on prices) However be aware that the traffic chaos surrounding New York is probably even worse than in the city proper.
800km is kinda simplifying it. For example if you hit traffic, if roads don't go the way you want them to. The region has a mountain range right there, and youd have to cross it to get into west virginia. Especially around major cities you could get stuck in traffic for HOURS going in and out of a city like DC if you are doing it at the wrong time. Its not gonna be as simple as driving down a street 800km. Now can you do it? Sure. Im just trying to make sure you know what your in for here lol. And hopefully if you do decide to go that route you can avoid some of those pitfalls. By doing things like parking at the outskirts of a city and using public transit to get around inside it. Altho i will say i've used DC transit before and its kind of a nightmare compared to somewhere like NYC. I tended to just use Uber there.
The main reason tho im saying a week isnt enough doesnt have anything to do with the travel distances between these places. Its that your just not gonna be able to fully experience each place you go before you have to leave. These are MASSIVE cities mostly, and your gonna be very time limited and miss out on some cool stuff in each city. If you think maybe youd want to ever do a trip like this again i would just do 1 city this time really see it in detail, and then do another spot the next time.
A good example of this is the statue of liberty. If you wanted to see just the statue of liberty it takes legit HOURS the ferries only leave at certain times of day, and your on the island for like awhile too. Just doing thats gonna take up an entire morning and probably well into the afternoon. That is just 1 thing to do in NYC. They have museums, cool stores you wont find elsewhere, parks, restaurants, etc. You could spend a month in NYC and still be seeing new stuff everyday.