I only mean theatre as drama.
You seem to mean theatre as physical buildings for drama production. "The theatre" also refers to drama, and you might also mean this.
Don't you have the puppet theatre where you live? If yes, how do you refer to it? We don't have specific theatres for puppets. As drama, it could be called "puppet theatre", though it could also be called "Cirque des marionettes". Neither the physical theatre, nor the type of theatre has to have a fixed name.
Do you have theatres for children? How are they called? Here they are called ordinary adult theatres that are used for children's drama productions.
Do you have ballet theatres? opera theatres? Yes. Examples are the Lyric Theatre and Performing Art Complex in Brisbane, the Suncorp Theatre (named after the sponsor); down south the Sydney Opera House (which is an opera theatre). Different drama and musical venues host various types of drama or musical production.
I understand that theatres have proper names, but if I want to take my daughter to, say, a puppet theatre, do I have to use the proper name or can I just say "We'll go to the puppet theatre tomorrow"?