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.
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.