It's hard to suggest a complete order without having played Pre-sequel yet. However, I would definitely start with the first game. Events in the second game assume you know a bit about the original Vault Hunters and some of the side characters. It's not vital, but I think it helps.
Multi-player is co-op.
Borderlands is about the loot. It's a game where your real goal is to get better weapons. I has an excellent sense of humor, good writing, memorable NPC characters, an awesome setting, and it's a lot of fun to play. The story isn't that strong in the first game but gets much better in the second game, to the point that I'm really interested in what's coming next. Handsome Jack, from the second game (and Pre-sequel) is one of the best video-game characters in recent memory.
Sirens are killer, Commandos are noob friendly.
There isn't much of the traditional RPG about, not like a Fallout type game, other than each class having three skill-trees with different play-styles.