Pretty much every innkeeper has rumors and "looking for work" dialog. They'll point you toward people to talk to or places to find things, or they'll give you a bounty on behalf of the jarl.
Wander around town and just talk to people. Some will just say hello, but others will have quests for you. Merchants, ships captains, townspeople, you name it.
If you're starting out, go to Farengar, the Jarl's mage in Dragonsreach. Ask him if he's the only mage in town, and he'll give you something to deliver to Arcadia's cauldron. You'll get some
free potions, and a large number of her items will change from "Steal" to "Take" after that, which helps greatly if you can't find that one pesky ingredient.
Off the top of my head, in Whiterun itself, Adrienne will ask you for a sword to be delivered to her father. The priest of Arkay in the Hall of the dead has a short quest to find something lost in
the catacombs. Carlotta (food stand) has a "get him to leave me alone" quest. Ysolda has a "bring me something" deal, which will open doors for another mini quest. Amren, the Redguard, needs his father's sword found. The guards will point you toward a bandit hideout across the river. There's also Danica Purespring, in the temple of Kynareth, with a quest, and, if you catch her at the right time, Mrs. Greymane has a pretty lucrative quest as well. I'm sure there are more that I'm forgetting right now, but each town has its collection of quest givers like these.
