Here's a common scenario - large video file uploaded to the site takes forever to load and might begin playing only after fully loaded. Is there anything we can do to make playback instanteneous?"
Every video file, besides having audio and video track, contains codec information on how to decode that video and audio data that's compressed to make final file size smaller. In H264 files (mov, mp4, etc.) that information is located in "moov atom". "Moov atom" can be placed at the beginning or end of the file, depending on the type of encoding tool used. If "moov atom" is placed at the end of the file, entire file has to be downloaded prior to being decoded and played; conversely, if "moov atom" is at the beginning of the file, video player knows right away exactly which codec to use and begins playback without waiting for complete download.
Incorrectly placed "moov atom" can be moved to beginning of the file using qt-faststart.exe utility.
Command: qt-faststart [SOURCES_FILES] [NEW_FILES]
There are also much simpler situations, with oversized and super-high quality video playing in much smaller video player, but that's the easy part - you just have to find an acceptable compromise between final video quality and resulting file size before publishing.