I was wondering what other forms are there besides minuet and trio, sonata allegro, and slow movement form? Also, what other types of pieces are there (i.e. String quartet, symphony, piano trio)?
Form is a thing that is tough to pin down. There are quite a few established classical forms – sonata, concerto, rondo, minuet/trio, large ternary form. But each one of these forms can take on an infinite variety of manipulation to where instead of looking for new forms, look at how the great composers made the forms their own.
The older baroque dance types and fugues are more a genre to me than a form per se.
Same goes with pieces, there are many standard ensembles like the string quartet, symphony, brass quintet, woodwind quintet, chamber symphonies, and many other non-standard ensembles as well.
Listening to a great variety of composers and more of their works beyond just their most famous can start to uncover these other pieces.
You can start by exploring http://imslp.org/wiki/IMSLP:View_Genres/Work_Types and http://imslp.org/wiki/IMSLP:View_Genres/Instrumentation