European Trip Suggestions

I love Prague but it is a total mob scene in the summer. Vienna is crowded but not packed like Prague and I have been to both cities in June. I haven’t been to Budapest so I can’t speak to that.

Without knowing too much about your interests or what you want to see/do, timing, etc beyond your mentioning keeping it to three cities, I think your instinct to consider Vienna, Prague, and Budapest is a good one. They're all quite close to one another, so you wouldn't lose too much time traveling from one to the other. Also, despite being so close, they're each so unique and distinct from one another that you're able to experience so much new stuff as you travel between them. I've combined those 3 cities on a few trips already, occasionally renting cars to drive myself or take trains, and occasionally throwing in additional destinations when on the trip (Brno, Bratislava, Dresden and such) when I wanted to mix things up a bit, but if you stick to Prague, Vienna, and Budapest you'd be doing just fine!

Prague is a mob scene not in summer too. I was there in April a couple years ago and was shocked by all the people. If you go, I would suggest staying in the side of the river where the castle is. We just happened to stay there and it was much more pleasant. We went to communist museum which was fascinating and I would highly recommend.
Vienna in contrast we found far less crowded. But it is more expensive. We ate really well for little money in Prague!
We took a train from Vienna to Prague and it was easy and pleasant.

Consider Poland, Summer time is a good time to visit and its four major cities Gdansk, Warsaw, Krakow, and Wroclaw each give a different taste of the country, you can always add Prague, Vienna, or Budapest with a cheap flight from Krakow or Warsaw

How about Greece? Athens and a couple of islands would be a great vacation. I would go as early in summer as possible.

I always wonder why so many people on this forum complain about crowds in Prague. I don't have that experience. What's crowded is so called Golden Royal Way connecting Mustek, Old Town Square, Charles Bridge, Castle. Everything else is like any other city. Of course you want to see Golden Royal Way. To see it without crowds get up early. Crowds will arrive by 9 am. I think crowds are much worse in let's say Venice, Florence, certain parts of Rome.

We were considering Prague, Vienna and Budapest. Any thoughts or
reviews of those destinations or other recommendations? We would like
to keep it to at the most three cities but could possibly do four.

The obvious fourth city is either Berlin or Dresden, which are both linked with Prague by a direct train. Berlin is a huge metropolis and deserves at least 5 days, with a day trip to Potsdam. Dresden is a great base for day trips to castles, mountains and cute towns, and offers many world-class museums and a small, reconstructed old town.

With 3 weeks, you could do Berlin, Dresden (it's great as an overnight, but worth stopping for a day on the way), Prague, Vienna and Budapest. I don't think I'd want a week in Vienna or Prague, but a week in Berlin and/or Budapest wouldn't be too much for me.

Or you could go to Ireland and Scotland, either avoiding the crowds of the Edinburgh festival or embracing the event.