It's definitely one of the must see attractions on your trip to Vegas. It's so large and central, that it's probably going to hard to miss out. My favorite spot is right at the center, if you get there five to ten minutes early, you will have a nice viewing spot.
As a little trivia, if you would like to pay over $250k, you will get a front row seat, a 30 liter Ace of Spades champagne bottle (look it up, giant is an understatement, it's an equivalent of 40 regular bottles of champagne), and a magical red button that when pressed will turn on the majestic spectacle that is the Fountains of Bellagio.
That said, I wouldn't consider staying at the Bellagio just for the fountains, it's rather a dilapidated and overpriced. You're much better off staying somewhere else on the strip and giving the fountains a quick visit.
I would check the schedule online, but here is rough idea:
Monday - Friday
3:00 p.m. - 8:00 p.m. show every 1/2 hour 8:00 p.m. - 12:00 a.m. show every 15 minutes
Saturdays & Holidays
12:00 p.m. - 8:00 p.m. show every 1/2 hour 8:00 p.m. - 12:00 a.m. show every 15 minutes
Sundays
11:00 a.m. - 7:00 p.m. show every 1/2 hour 7:00 p.m. - 12:00 a.m. show every 15 minutes