The way that the audio and video are synchronized for film, TV and the like is that each is recorded using a marking system called
SMPTE Time Code which basically puts a mark on each video frame that corresponds with on mark on the tape, allowing each to be connected during editing.

It is helpful to understand that much of the music we see on film and television is recorded either beforehand, in the case of musical numbers, and lip-synced, or afterwards, as in the case of background music. Also helpful to understand that when events are filmed, there generally is a camera shooting silent, asynchronous footage that can be edited into the master footage.

As to your project, there are ways to do it without having recorded using SMPTE, but there will be a lot of tedious, imprecise, cut and paste.