I started many campaigns in taverns, but now, I don't have a go-to. I try to tailor campaign kickoffs to the party in such a way as to let them get a feel for what is to follow.
There is one method I've used with increasing frequency: each PC gets to star in a mini-scenario that all eventually tie together. The first time I did it, the thief was doing a little 2nd story work when he triggered a trapdoor that dropped him into the sewers.
...basically on top of a kidnapping in progress (victim: the wizard). The kidappers, caught off guard, turned on the thief. But the wizard woke up in time to take down one kidnapper, and the thief got the other. Together, they found their way out of the sewers.
...unfortunately, right near the people who hired the kidnappers. Chase ensues, and the fleeing duo literally run into a couple more PCs who are having a discussion near a foodseller's stand. The thief, who can practically smell do-gooders, fast-talks them into helping thwart the attempted kidnapping.
A true breach of the peace ensues. At some point, a cooking fire gets toppled when the barbarian (literally) got tossed out of a bar, and The Watch and firemen are called. Everybody legged it towards the wharf as the fire spread and the Watch showed up.
When the all-clear eventually sounded, the PCs- who have been talking to each other, trying to figure out what happened- wandered back to the square to find breakfast. The "do-gooders" droped their meat on a stick, etc., when they realized the pile of ashes before them was the inn they had each been lodged. They had only what they had been carried- everything else was lost in the fire.
At that point, a merchant approached, and offered that twosome a chance to get back on their feet as caravan guards. They accepted, and used a bit of "you owe us" staring to get the thief to join (giving them part of his salary). The wizard, fearing another abduction attempt, volunteered. The barbarian, sobering, sensed an opportunity to make more beer money...