Here are some points for the reference if attend concerts

June is a bit tricky as it is close to the end of 23-24 season. Check online for the concert schedules

1. Berlin

BPO at Philharmonie Berlin (Try to buy ticket well in advance)

Either Staatsoper Berlin Unter den Linden or Deutsche Oper Berlin

Check Konzerthaus Berlin. Sometimes it has high qaulity visiting performance

2. Leipzig

The famous Gewandhaus Orchestra (try to buy ticket in advance)

3. Prague

If Prague Spring festival is still going on, check its website to get a ticket to performances. Otherwise check Czech Philharmonic schedule. Czech Phil normally performs at Rudolfinum concert hall. 

Try to watch an opera in 1 of 3 Prague opera houses. Tickets are so cheap compared to US.

4. Viena

Of course the famous VPO and Vienna State Opera if they still perform in June.  Get ticket online well in advance.

5. Salzburg

The famouse Salzburg Festival is in late July and Aug. In June, check schedule in Mozarteum for authentic local Mozart concerts. 

I strongly recommend to stop by Dresden to get an ticket in Semperoper Dresden if an opera is available in June. Dresden is on the way to Prague from Leipzig. Semperoper Dresden is arguably the best opera house in Germany.

In all of these cities, there are many concerts designed for the tourists. If you are a serious classical music fan, try to avoid them.