The Fountains of Bellagio provide one of the most sought-after views in Las Vegas and when Steve Wynn opened the place, he was smart enough to locate most of the restaurants directly on the water. The best view among the great views is probably Le Cirque, dead-center on the second floor. It's also a fairly small dining room; hence, quite a few tables have primo views. But Picasso, Prime, and Osteria del Circo all have nice views of the fountains.
Ironically, perhaps, Paris across the street offers the best aerial view of the fountains you'll find in a restaurant. For that high-altitude perspective, nothing beats the Eiffel Tower Restaurant on the 11th floor; reserve early to score a window seat. But no matter where you sit, this is also one of the city's most romantic restaurants.
For great views of the city as a whole, there's Top of the World at the Stratosphere. The revolving restaurant (360 degrees every 80 minutes) 800 feet high has undergone a major upgrade in food over the past few years and is now serving cuisine that matches the happy hour bargains. Just don't let the thrill riders plunging past your window or dangling off the edge freak you out!
Nove Italiano atop the Palms' Fantasy Tower also has a great aerial view of the Strip, thanks to the resort's unique position a few blocks to the west. Almost every seat here is amazing. But the ones against the east wall have the best panoramic vantage point.
Mon Ami Gabi in Paris offers casual French bistro fare and if you sit on the patio, you can people-watch on the Strip and watch the Bellagio fountains across the street (though the exhaust fumes from traffic at the busy intersection of the Strip and Flamingo sometimes get a little overpowering).
A couple good restaurant-with-a-view options are found at The Cosmopolitan, especially the patios at Comme Ça and Estiatorio Milos. The former looks out onto the Eiffel Tower and the north end of the Strip, while the latter has a great view of CityCenter, Planet Hollywood, and points south.
For an altogether different type of view, try Aureole in Mandalay Bay. The main attraction here is the wine tower in the main dining room, in which Aureole's signature Wine Angels "fly" up and down to retrieve your bottle. Their outdoor Swan Court offers a view of a different kind: Your table, on a private patio, sits on a small lake.
Out in the 'burbs, Marche Bacchus has amazing patio seating on the man-made Desert Shores lake. Feed the ducks and black swans while you eat or just enjoy some rare waterfront dining in the desert! The over/under on the comment, "I can't believe I'm in Las Vegas," is five minutes (take the under).
As far as getting a good table's concerned, it's always best to request it when making a reservation. (Telling them it's a special occasion might help.) Many restaurants will now let you reserve a window table for a fee, which could range anywhere from $10 to $100, depending on the restaurant (and the view). This is essentially an extra fee to hold your spot and ensure a sweeping view of the strip!