You start out at the Merchant Square in Gilneas City. Talk to Prince Liam Greymane, who is the person on the horse right in front of you. He has a yellow exclamation mark on his head, indicating he is a quest giver and has a quest for you.

Lockdown
The prince tells you the city is under a complete lockdown, and that you should go see Lieutenant Walden for further evacuation instructions.

Quest Giver: Prince Liam Greymane
Location: Gilneas City Merchant Square
Recommended Level: 1-3
Quest Objectives: Find Lieutenant Walden in the NW end of Merchant Square
Rewards: 15 copper, 80 experience

You should see a yellow question mark on the minimap. This denotes a quest objective. If you don't see it, press M to bring up the map.

Head Northwest towards that question mark. Follow the road and you will find Lieutenatnt Walden's Corpse near the City Gate. Right click on him to complete the quest. Congratulations, you have completed your first quest!

You now receive a new quest

Something's Amiss
The Lieutenant is dead. You should let the Prince know.

Quest Giver: Lieutenatnt Walden (corpse)
Location: Gilneas City Merchant Square
Recommended Level: 1-3
Quest Objectives: Deliver the news to Prince Liam Greymane in the Merchant Square.
Rewards: 35 copper, 170 experience

Head back to the starting point and talk to Prince Greymane again to complete this quest. He will offer you two new quests, accept both of them.

All Hell Breaks Loose
The Rampaging Worgen are attacking. Help with the defense.

Quest Giver: Prince Liam Greymane
Location: Gilneas City Merchant Square
Recommended Level: 1-3
Quest Objectives: Slay 6 Rampaging Worgen
Rewards: 65 copper, 220 experience, Wolf Fur Coat (cloak, Bind on Pickup, 6 armor)

Slay the Rampaging Worgen that are attacking the Prince. They are not hard to kill. As you kill them, you should see a number show up showing how many you killed.

Kill at least 6 of them to complete the quest. This should also level your character up to level 2. Congratulations!

Evacuate the Merchant Square
Civilians aren't safe any more. Help evacuate them.

Quest Giver: Prince Liam Greymane
Location: Gilneas City Merchant Square
Recommended Level: 2-4
Quest Objectives: Evacuate 3 Civilian Homes
Rewards: 65 copper, 220 experience

Bring up the Map ("M") to find the locations of the homes to be evacuated. Look for doors that have a sparkling effect in front of them.

Report back to the prince once you've finished the evacuation.

Salvage the Supplies
Supplies are getting destroyed, and you must do something about it.

Quest Giver: Gwen Armstead
Location: Gilneas City Merchant Square
Recommended Level: 2-4
Quest Objectives: Salvage 4 Supply Crates
Rewards: 65 copper, 220 experience, Salvager's Bag (Bind on Pickup, 6 slot bag)

Look for supply crates with shiny/sparkle effects. Report back to Gwen once you've found four crates.

Royal Orders
You are needed at the Military District

Quest Giver: Prince Liam Greymane
Location: Gilneas City Merchant Square
Recommended Level: 2-4
Quest Objectives: Head to the Military District to the Southeast, and talk to Gwen Armstead
Rewards: 50 copper, 250 experience

Use your map ("M") to make way to the Military District to the SE, and locate Gwen Armstead down the stairs and across the bridge.

You should now be at level 3 (or very close to it, if you are not just kill some more Worgen)

Gwen gives you a quest that sends you to your class trainer. This quest will be different for everyone, depending on your class. Follow her instructions and locate the person whom she sends you to meet. You will most likely get a new ability and a quest that will let you use that ability. Learn the ability from the class trainer, and do the quest he sends you on.