Berlin can boast three opera companies that will be beginning their seasons in September. The State opera calendar is here http://www.staatsoper-berlin.org/en_EN/calendar Note that its old home is under renovation and it is performing in the small Schiller theatre which is marvellously intimate; the orchestra has to file into the pit through the auditorium. The Deutsch Oper Berlin is located in the same area in a post-WWII theatre (opened 1961). http://www.deutscheoperberlin.de/en_EN#
The Komische opera in the former East Berlin specializes in German-language productions http://www.komische-oper-berlin.com/
These facilities play host to recitals, concerts and even ballet, and there are plenty of "alternate" performing enterprises, so Berlin is truly rich in classical music.
Tickets prices can be very moderate by North American standards, thanks to government subsidies. On-line sales are offered, but late last September I bought a good ticket from a government tourist office at the main train station for two nights later.
If this might be your first exposure to opera, keep in mind that most opera houses these days run "surtitles" translating the sung language -- but, being Germany, those translations are going to be into German.