I am a pretty serious handgun hunter, so I will throw in a few thoughts. Single action revolvers have a different feel to them compared to double action revolvers, so try to handle a couple of each, and see what feels better in your hand. SA's roll in your hand with recoil, DA's typically recoil more straight back into your palm. Longer barrels are popular, but can be a pain to carry. If you are using optics, then a shorter, say 4 to 6 inch barrel is easier to pack, gives up little in ballistic performance. The longer barrel sighting plane increase is not a factor with optics. You can pick up a new S&W 629 or Ruger Redhawk / Super Blackhawk for right at about $1,000, or less, depending on model. A decent red dot, like a Sig Romeo or Similar Vortex or Ultra Dot, about $150. You can spend more, but these will hold up to recoil fine.
Whatever you choose, make sure it's drilled and tapped for scope mounts, or has rib cut outs like some Rugers do - Makes mounting optics a lot easier.
Caliber? I have taken deer with the .357, but find a bigger bore like the .44 mag or handloaded .45c hits harder, and tends to drop them quicker. I like the .44 mag loaded with 240 grain XTP bullets. Good expansion, and plenty of penetration for whitetails. Hard cast bullets with a big meplat also hit hard, and typically penetrate deeper than an expanding bullet, but unless you hit the CNS, expanding bullets seem to drop them a little quicker, especially on heart/lung shots.
Red dots are quick and accurate, magnified scopes can help place your bullets better in tricky shots, but some people have a hard time finding their target, and are put off by the scope magnifying your shakes/movement. Try looking thru guns with each type and see what works better for you. If you are going with a magnified scope, 2x is fine.
My personal favorite big game handguns are a Redhawk with an Ultradot sight, a S&W 629 with a Sig Red dot, and a S&W 629 and Super Blackhawk with irons. Accuracy wise, my standard is being able to keep all six rounds on a dinner sized paper plate from a field shooting position. Usually 50 to 100 yards for me, depending on what gun/sighting combo I use.
Larry