Interesting journey! You are going to have a helluva trip.
Personally, my preference would be to go south from Budapest to Zagreb, Sarajevo, Macedonia, and into Greece from there, rather than heading east to Bucharest and Sofia. Especially if this is your first time traveling in the Balkans. It gets sketchy east of Budapest, pretty quickly. But Bosnia and Croatia have big tourist seasons and are more used to accommodating (and protecting) travelers.
As you head east, it becomes more of a manly-man "every man for himself" kind of environment, where the police are more likely to laugh at your mugging, rather than help. It takes a certain type of traveler to thrive in Romania and Bulgaria, and if this is your first time, it might be better to go to the more well-traveled routes along the Croatian coast, for example.
Cars vs Trains
When I moved to Budapest eons ago, I too had been raised in the car-owning world of America, and couldn't imagine how trains could adequately replace cars.
Then, when I moved back to America seven years later, I couldn't believe how backwards and fucked-up the American car culture is. It is SO MUCH EASIER to get around Europe on a train, than to get anywhere in America in a car.
"Shit. I missed the train to Novi Sad. Well... let's see... there's a train leaving in 30 minutes for Sekesfehervar, and from there I can go directly to Zagreb, and it'll only be an hour longer. Ok! Cool! Let's take that one!"
Be forewarned, Greece can be tough. Not much of it looks like the Greece we see in pictures. Athens is a big city with a lot of smog and scorching heat, at times. Almost all the Greek islands are deserts.
Delos will blow your goddamn mind. You take a boat out there, and walk around an actual Greek city that people just abandoned about 1500 years ago. Streets, buildings, statuary... it is the closest you will ever feel to what it was like being a Greek in the heyday of the culture.
Santorini is ... everything. Utterly enchanting. Unreal. Like an epic movie set. And yet, if you get down to the south end of the island (Kamari and Akrotiri) you will see ancient Greece right there on the surface. There is an old amphitheater in Akrotiri at the very top of the hill, and it will blow you away.
Lots of people go to Mykonos. If drinking and getting puked on by badly behaved foreigners is your idea of a good time, definitely go to Mykonos.
When going back north into Europe, you should look at both Slovenia and Trieste, in Italy. Maybe a boat up the Croatian coast from Greece. Slovenia is gorgeous, and is basically a cheaper version of Austria with nicer people. (Not that Austrians aren't nice... but they have that Germanic/Teutonic coldness about them, while the Slovenes are Mediterranean types.)
Whenever anyone goes to Vienna, I always tell them to go to the Vienna Clock Museum. It really is special... the building, the collection... even if you think you don't care about clocks, it is a lovely place. And right in town, too, so it is easy to visit.
Big Picture Stuff
Pack one bag. Carry it around with you everywhere for a month before you go, to get some practice handling it deftly. And no matter how much stuff you leave with, you will always come home with one bag. Start off that way, and save yourself the hassle.
We think we need way more STUFF than we actually need. And you will shed about 50% of the shit you brought in the first week. The second week the rest will get stolen. (Yes, it will.) So you will replace the essential items with (invariably better) items you find along the way.
"This shirt? Yeah. I got it in Sarajevo when my stuff was stolen on the train in Serbia."
Bring a couple of simple items of clothing. Buy stuff that is culture and weather-appropriate once you get there. This will help you not stand out like a goddamn tourist (and decrease your chances of getting robbed), and will also force you to interact with the locals when you look for, and buy, the things you discover you need on the road.
Write. Even if you aren't a writer now, make a point of writing every single day. Take notes. Town names. Sights. Train rides. Experiences. SO MUCH will happen to you, and your mind will go through some amazing changes and transformations. Having daily notes will be TOTALLY FASCINATING when you get back.
Back in America
Find people to connect with who have traveled recently, BEFORE you leave.
When you get back, you are going to be super disoriented. It is HARD to talk to Americans about the rest of the world and anything that isn't on TV. You will have a MUCH easier time acclimating back to life in America if you can go hang out and have dinner a few times with traveler friends.
It took me FIVE YEARS of totally floundering around before I found my feet again in America. And I was only gone for 7 years.
Your life is about to change, and you are about to become a citizen of the world. Welcome! It'll be nice to have you here.