The fountains put on their show -- wonderful timed to music choreography -- every half hour on the half hour, starting at 3PM M-F and 12 Noon  on weekends.  The shows last until 12 Midnight (Note: show is every 15 Min from 8PM to 12 Midnight).  The show lasts for a few minutes (length varies according to the music). It's very pretty but very brief. However, there's plenty to see at the Bellagio, so time your visit so that you can pick up at least two fountain displays.

It will be a good idea to stand at the pavement next to the main road and facing the hotel.  At this location, you would have a better view and perhaps good for video taping the show.  However, the driest location is to film from the pedestrian bridge that crosses the Strip from Bally's to Bellagio.  You can hear the music from there, get a more inclusive field of view without traffic interference, and not be at risk of becoming damp from the water flumes.  Or, for a whole new take on the show, watch it from the top of the Eiffel Tower across the street at Paris.