This game was insanely popular in the nineties, and the rules of the game are simple enough to understand. The challenge to running a successful Vampire game is getting a feel for tragic roleplaying. Vampires are not heroes, and they are not all powerful. They are eternally damned, and being a vampire, for lack of a better word, sucks.
Many of us have years of experience, and we have great stories to share, which could be reused to great effect. The political intrigue is the highest of any game ever, and there is an unparalleled capacity for interpersonal dynamics. Essentially the players form a coterie, and are a group of vampires who, for whatever reason must live and work together for their continued survival. You can't meet in a bar like most dungeon crawlers start, the all of you are more like an unholy family.
As the game is formed, each player will have to choose a clan. People who have played before should defer to the newbies to let them pick first clans, and entice experienced vampire players to try a new one (though in this game, people are serious about their favorites.) Give them a couple weeks to really think about their character backgrounds, and try to tie the group together as tightly as possible, because nearly every circumstance in V:tM is devisive.
As storyteller, resist the urge to create your own character to play along side the players. Babysitting is a bad idea in any game, and as the players learn the basics of hunting and feeding (there are several ways they can) the game will run itself as you enforce the Camarilla code on the players until they understand the need for secrecy and respect for authority.
Nature/Demeanor are the best mechanics for awarding points for roleplaying, and while it may not be so essential in a Steve Jackson game, White Wolf thrives on good roleplaying, and without it, is a very shallow system. Get people to take the game seriously, and it can be one of the most rewarding table top experiences imaginable.
Start the players off near 13th generation. You'll want to explore every aspect of a vampire delimna in a rising action setting, first is rumors and gossip, then a holy man starts making noise, it will culminate in a mob mentality before the characters learn that they will never rule the day. Vampire hunters make excellent recurring villains, and from there you can move on to bitefights with other coteries, or werewolves, or the many many mysteries in the World of Darkness.
Get ready to make some NPCs. You will introduce a character for a necessary plot point, and end up using and reusing the same characters as the story demands, but don't get too attached to them either, as the players will likely solve most problems with violence - it's just hard coded into most gamers. Under no circumstances should you let the players use diablarie (sp?) as a form of powerleveling. Anyone with Auspex can immediatly recognise a diablarist, and no Camarilla will suffer one to live.
No game has better supplements than Vampire. Do not miss out on the excellent stories of the 'By Night' series (New Orleans is my favorite) and any one of the adventure modules could serve as a setting for an entire campaign. Run it into the ground and then run them out of town.
There's a lot more could say, but I'd like to hear about your expectations, and more about your gaming group. I'm sure there are plenty more people going to post as I have as a bit of nostalgia. It is one of the most immersive roleplaying settings imaginable, and while biting each others necks may not seem titillating to you now, it soon will.