If you want to follow the story all the way through, your best bet is to play them in chronological timeline order.
Resident Evil: Zero
Resident Evil/Resident Evil: Director's Cut or even the Remake
Resident Evil 2
Resident Evil 3: Nemesis
Resident Evil: Code Veronica/Code Veronica X
Resident Evil 5
You'll notice I didn't put Resident Evil 4 up there and that's for the simple reason that it's, essentially, an unrelated game story wise. If you wish to play this game, my only suggestion is that you play it after you play Resident Evil 2.
The Umbrella Chronicles and The Darkside Chronicles just touch on the games and the timeline and actually, in my opinion, kind of butcher the story and the entire feel of Resident Evil so I wouldn't recommend playing them until after you've learned the story by way of the other games.
If you wish to branch out into the Survivor and Outbreak series', you should probably play Survivor somewhere between RE2 and Code Veronica since it makes references to Leon but happens before Resident Evil 4. Gun Survivor: Code Veronica should be played after you play Code Veronica and Resident Evil: Dead Aim would be best appreciated if played after Code Veronica as well.
As for the Outbreak games, I would say play them anywhere between playing RE2 and RE3 because, even though they're unrelated to the story of the main series, they still deal with Raccoon City so it's probably best to get all the Raccoon City era stuff out of the way before moving on to different areas of the Resident Evil universe. (Unless you wish to finish the main timeline before branching out into different, unrelated games. Once you get into Survivor and Outbreak, it's becomes less about storyline and more about personal preference.)
Hope this helped and welcome to the Resident Evil universe!