A $500 budget basically opens many options for you. My recommendation is -
1) Buy Uberti - many will argue over a quality difference but Uberti has the edge and you can't argue with the aesthetics given the fact that Uberti hides the ugly Black Powder disclaimers whereas Pietta puts them prominently on the barrel. Just ugly and stupid to put this info so prominently on a replica.
2) Buy 44 caliber - more fun to shoot a larger load, which you can also reduce if required. Also, when you want another gun - which just happens once you get into BP, your 44 equipment, balls, etc. can be shared as there are many 44 caliber options to make owning, buying, shooting, and enjoying additional guns easier.
3) Buy a Colt replica - The history and evolution of the revolver are in your hand when you shoot a colt.
4) Buy a replica of an actual historic gun, caliber, and barrel length - an actual historical shooting experience is just plain enjoyable
5) Avoid brass frame options. Your budget eliminates the requirement to skimp and buy a lesser quality gun.
6) Consider replacement nipples for your gun from the start in your budget - Ampco, slick shot (my preference), etc. More colt replica new and used options pop up if you watch and you need to factor in the equipment needed to shoot and clean.
7) Start looking for percussion caps after you read this and buy some when found regardless of price so you can shoot your gun when you get it instead of just fondling it for a while. BP is available so less of an issue right now.
1st choice recommendation - Uberti 1860 Army 44
2nd choice - Uberti 2nd Generation Dragoon 44 - big guns are really fun to shoot and make for great conversation at the range. Suggesting 2nd generation only because that was the version that originally sold the most but a deal on any model Dragoon is worth it.
3rd choice - If 44 is not the way to go for you then the Uberti 1851/1861Navy as others have noted.
Enjoy your BP journey.