You start your druid specific quest journey with the standard 1st-level "take this (object) to (early trainer)" quest. This quest is given when you complete the very first quest you are offered upon entering the game. You get this quest from the NPC standing about 10 feet in front of you when the cinematic ends. When you turn in the first quest, the "go visit your trainer" quest will be offered to you. This quest shows a new player where the druid trainer is located in the starting area, so the quest description always has all the information you need.

Starting at Level 10, more in-depth class-specific quests are available. All of these quests start with your 'quest trainer', who is a normal class trainer but who also gives out the class related quest starts. There are "leader" quests to direct you to this person if you are in another city when you reach the appropriate level.

The Alliance 'quest trainer' for Druids is Mathrengyl Bearwalker in the Cenarion Enclave, in Darnassus, Teldrassil. He is on the second floor of the Druid tree. Ask a guard for Class Trainer → Druid and they will point you the way to the tree.

The Horde "quest trainer" is Turak Runetotem on the Elder Rise in Thunder Bluff, Mulgore. Again ask a guard for Class Trainer → Druid.

All these quests are multi-part and some parts have multiple steps. These are outlines of the whole process to accomplish the series of quests.