I own every gun game on the system, and yes I mean everyone. (Even import ones)
But as far as US releases, I really like Endgame, and RE: Dead Aim, I like the idea of playing a RE game with FPS-like aiming, that I control.
Dino Stalker is frantic, prepare for a tired finger but its kinda like a FPS (walking around and trying to not get back attacked in some areas is next to impossible) meets gun game approach makes it harder than it needed to be, still its fun, but look into it before plunking down the cash. It can be played without the gun like Dead Aim, but it just isn't as fun.
I recommend Time Crisis 2 & 3, but they can get boring fast, at least 2 has a target shooting area, which more games should have in them. I will recommend Vampire Night to a lesser degree but you have to be too accurate in that game so it lessens the fun, but at least it's nice looking.
Oops forgot one, Starsky and Hutch, it's not the greatest game ever made, but it makes for a really cool co-op experience, one person drives and the other one shoots, even though the gun's calibration gets out of whack in the game occasionally, my brother and me had fun playing it a lot, he drove and I shot, and it can be had dirt cheap, so if that's something that interests you then look into it.
I don't like Time Crisis: Crisis Zone or Ninja Assault (it looks, plays and feels last gen) if you can play imports some guy here on CAG was selling the Gunvari Collection, which has 4 PS1 guncon games compatible on the PS2 and Guncon 2, but like I said it was import only.
Another import I would recommend is Gunfigher 2 The Legend of Jessie James (I think that's its subtitle) the only issue with it is that it's PAL only, but it was made by Rebellion, and for a old west gunfighter, I enjoyed it quite a bit.
That should get you started.