Well tickets in July or August are going to be at their max price pretty much. June would be alot cheaper, if you can swing that.
You should look into many different places in these countries. Amsterdam is a given in the Netherlands, but why not check out Haarlem as a day trip? It’s barely 20 min by train away.

For Germany, consider Berlin as well as Munich. Both have alot of history and interesting spots. Munich may better suit your course of travel between Italy and Holland though, as Berlin is sort of “out of the way”.

For Italy, Rome is the “main” place people usually go, and why not considering the ancient history there not to mention the Vatican, but Florence is huge too for the art and renaissance history. Venice is big because of how unique it is, with the buildings propped up on stilts and canals as roadways. What kind of interests do you have? That might help you decide for sure on a place.

Start in the Netherlands, flights from the US to Amsterdam are going to almost exclusively be cheaper to Amsterdam than anywhere in Italy, adleast that is the case from my major city (Atlanta) then you could take a cheap flight from rome to amsterdam before you return trip to america. in terms of cities, i have heard that rome is only good if you spend a lot of time in it, florence is a good alternative for a shorter stay or a more artsy trip, naples or genoa for an overload of culture.