I've always liked the fact that shotguns are a very personal item and as such what one person considers the "best" doesn't work for another shooter. What follows are my choices. They might not work for you.
For under $500, the Remington 1100 but a well-cared for older model. The latest ones aren't very impressive.
At $1,000, another vote for the Beretta 390(1).
$2,000 gets you into a quality over/under. I don't care for the Browning 12 gauges (the 20s are nice though) and am firmly a Beretta shooter. One of the 680 series guns is my choice.
And after that the world gets cloudy. I've hit the first three stages and taken a half step in to the fourth (a well used Krieghoff Model 32 that set me back about the same amount as a Browning). This is where the decisions get harder. Krieghoffs are everything that Geoff says they are. Mine reminds me of driving my uncle's 1970's Buick Roadmaster. Big, solid and smooth, low recoiling.
Perazzis are another very popular choice but they are like driving a sports car. Nimble, quick and great handling. Which do you prefer?
Kolars are gaining a deserved reputation for quality and the Spanish-made Kemen is turning some heads now that they are being regularly imported into the USA.
If the absolute sky were the limit I might choose a Fabbri pushing close to $75,000 or a Galazan o/u which starts at $38,000 without engraving. (Matching the engraving on a Fabbri puts the Galazan into the same rarefied neighborhood.)
So many shotguns, so little money.