In short: 1, 2, 3, 4, and Code: Veronica.
The older ones (1, 2, 3, and Code: Veronica) are puzzle-oriented tank controlled, fixed camera survival horror games. I love them. Though if you want play the first one, see if you can get the Gamecube remake. Better in every regard.
Zero is done in the older style, but has a two-player focus (one is AI, and you co-operate with them). A fun game, and if you like the series, go for it.
Outbreak and Outbreak: File #2 are fun games, same idea as the earlier ones, but are done in short missions and designed for online play. Since the servers are disconnected, this is not possible. The AI is terrible, so you have to be DAMNED good to finish them. Even then, that might not be possible.
RE4 is great, though it deviates from the original style - more tense than horror, has an over-the-shoulder perspective, and much more emphasis on weapon upgrades, with less on puzzles. A great game. It was a good reboot for the series.
Which brings us to 5. Pretty much an imitation of 4, but not nearly as good. (I dislike it quite a bit). If you want to play it, do it with a friend. The single player is frustrating, I'm told.
Survivor is a first person, arcade-style game. Not very good. Don't bother.
Dead Aim is a spin off, and more like RE4. A decent game, but very easy (ammo is unlimited). It switches between third person, over-the-shoulder perspective, and goes into first person for shooting. Very few puzzles.
Umbrella and Darkside Chronicles are first person rail shooters for the Wii. They pretty much just tie up lose ends in the series. Not that great. If you love the series, you might want to play them.
I haven't played Mercenaries 3D, Gaiden, Survivor 2, or Deadly Silence.
Hope that helped!