Visited the States Boston (4days) and New York (8days) last summer for the first time from Europe as well. Canīt reccomend any place for tickets but I will give you some reommodations for what you can visit.
, I would reccomend to walk the "Freedom Trial" its a path through the city and takes you along the most historical buildings of Boston and is an easy way to get a free city tour by walk.
You can check the Bill Russel statue which is placed next to the Boston City Hall and there are some things to explore in the area around.
You can visit Harvard (free)
Get on top of the Prudential Tower (Def. worth the money)
Go Shopping at Wrentham Premium Outlet (Depending on your Budget, and I donīt know how much clothes cost in Hungary, but for me as a German it was a great deal. I was allowed to bring back stuff for 450 and I basicly spend all of it in a wrentham outlet in Philly(and I realy was on a Budget but it was just so much cheaper compared to Germany especialy Calvin Klein, Tommy Hilfiger) In Boston as in Philly itīs 1,5h or something outside of the city centre so you have to check trains/busses or rent a car but if you want to do some shopping it will be worth the way.
In New York.
Spend some hours in the Central park (really loved the Central Park!)
Get on top of the Rockefeller Center (other option is Empire State Building, I rec. Rockefeller)(Must do!)
Visit Wall Street/Ground Zero/Downtown and take a Fery to the Statue of Liberty (free Fery with Metroticket)
Visit the Time Square(Must do!)
Walk over the Brooklyn Bridge(Must do!)
Just walking around in Midtown/Downtown of Manhattin is pretty Amazing
If you want to go out in the evening, the "hip" parts of Brooklyn is the place to be. (Nightlife in the US sucks(no offense guys
) compared to Europe, but I guess your not looking for that anyways)
If you feel like you have a day left to spend, you can make a 1 day trip to Philly by Bus (traveling by Bus is very cheap(Megabus, Greyhound))
In both Cities buy a 1 week Metro pass. Everything can be easily visited using subway and it is definitly worth the mobility for the price.
Genral Advice: If there are no prices indicated, it is most likely expensive. Consider that there are no taxes included in the shown prices. Waiters will expect decent Tipping.
Have a great time, if you have further questions, feel free to ask