Decide what type of puppets you want to build. Many different types of puppets exist that work for puppet theaters. These start simple, such as foam hand puppets, and range to the extreme, such as the large puppets that are run by several people at once. Many other options in between exist.
Take workshops in puppet making. Many existing puppet theaters and puppet specialists offer workshops in puppet building. Look for one in your area. Work within the expertise of the teachers, but also use the type of puppets that you want to build as your inspiration in the class.
Find enough scripts to put together a season with. Either find existing puppet scripts through playwriting publishing companies and acquire the rights to them, or start writing your own. You can make a season with a single show, or as many as five or six depending on the scale you want to attempt in your operation in your inaugural year. It is best to start small and work your way up as you adjust to the learning curve of running the company.
Start booking gigs for your theater. Submit yourself for spots in children's festivals, fringe festivals and private gigs at schools or any other events where you feel you can market yourself.
Build your puppets, a portable stage if you require one, cast your shows and start the rehearsal process for your season. Remember that a traditional "puppet show" type stage is likely not appropriate for a puppet theater that you will show professionally, unless you only perform in very small venues. If you want to keep the performers concealed, something as simple as a very long and low curtain works well while providing you with lots of room to maneuver bigger puppets that are visible from a larger space.