Another U1 supporter
Many of the modules mentioned above are amoung my favorites to DM 1st lvl characters through, but the one that has consistently gotten the best remarks from my various
*players* over the years is:
Sinister Secret of Saltmarsh
MINOR SPOILERS
(tried to be careful though, not much more info than you'd gather from looking at the module cover)
It really has most of what you'd want in an introductory adventure. The first part is an exploration of a haunted mansion, which has a great "mood" to it (and an NPC which when done well by the DM can add more to the story). Players like the little "item" tidbits scattered around in the fine details of the house. The second part moves to a cave system where there is a good villain and more unveiling of the mysterious background plot (which can then lead to lots of DM-created role-playing back in town). Then the last part is a sneak attack onto a ship, which is a great "stage" for glorified combat, has some great villains aboard, and some interesting "neutral" NPCs for the DM to role-play. Then you end up with the option to continue the story onward with U2 (which promotes role-play to avoid disaster), and U3 (big combat).
It was written in the earliest days of module writing so it's got a different feel than recent ones, but the varying scenery, super-detailed maps, slowly expanding plotline, useful/unique NPCs, and "named" villains make it really worthwhile. Some last minute fine-tuning by the DM would probably be beneficial for 3E, but all in all it's a great choice. It used to be available as a free download from the old TSR site, and a 3E conversion was also recently available on Eric's old conversion page (but I think that page was just taken offline....).
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My other votes would be for many of the same things that everybody else already suggested (in rough order of preference):
Of Sound Mind Fun/original ideas, you can't go wrong with this one
Sunless Citadel Don't run it if that one player went through it already, but it must be included on my list anyway because it's a mandatory "right-of-passage" for all 3E players, it gives them a connection/conversation with any other gamers that they meet
Village of Hommlett (the original T1) Excellent background info for a campaign's "home base", probabably the best village details ever written, has tons of random new-plot generating tidbits everywhere, it has a decent first dungeon included but the whole module is really meant more as a seed for future plots (and it delivers seeds aplenty)
Gorgoldland's Gauntlet Great for newbies, fun/easy puzzles
Keep on Borderlands Excellent starting "town" details, unrealistic dungeon crawl but your players will have alot of fun playing it anyway
Under Illefarn I agree with BluWolf, it gives you all you need to start an extended campaign, has excellent background info and excellent starting "town" details, then adds some good introductory "mini-adventures" to get things rolling, the main dungeon is a bit dungeon-crawlish but everything leading up to it is great
Raiders of Galath's Roost First part of adventure is very good and intriguing (it has a typical dungeon feel to it but it adds some inspirational details), the second part is dangerously harder/combat-heavy (depending on how it's played)
Treasure Hunt I concur with Sigil's comments exactly, a bit of an unusual start but it's meant to be a first "intro/newby" type adventure anyway
Wizard's Amulet/Crucible of Freya A good choice, but I think this storyline ranks below the top-notch adventures in terms of encounter variety and role-play potential, however there are a couple redeeming highlights that make it all worthwhile and fun enough anyway
NeMoren's Vault Another good choice, can maybe be used as a second adventure after one of the shorter first ones
Tomb of Horrors Just kidding

But, I actually did a "toned-way-way-way-down" 1st level non-lethal conversion to scare some players to death once.... it worked and was
alot of fun -- besides, anything with "pictures" to show the players is always a winner, like the Kenzer series' picturebooks (however, there's virtually "no" roleplay in ToH anyway, so it's probably not a good choice for a first timer even it it
is sacrilegiously "de-clawed"

)