Is there a way I can look up flight plans filed in the United States? I tracked an aircraft on Flightradar24 and I would like to be able to find it's flight plan in some sort of database.
It seems you are looking for domestic flights in the US. The FAA database is not public.
You can use ADS-B tracking sites to look at the route. Remember instructions after takeoff (departure and transition) and before landing (transition and approach) are determined late and can be changed, depending on winds and other parameters.
E.g. on Flight Aware, for JBU124 between LAX and JFK, the route is mentioned:
and can be decoded:
These sites also create route history records. After a certain number of days, it's paying.