This page of The Last of Us game guide contains a walkthrough of the 20 years later stage as part of The Quarantine Zone chapter. This level does not yet involve any fighting or sneaking. Instead, you will learn how to explore the area and about the basic mechanics of the game, such as the usage of Med Bags.
After this chapter begins, keep following Tess. You can take your time here - it is a good idea that you eavesdrop on the citizens' conversation while on the way. Thanks to it, you will learn quite a lot about the conditions in the zone. The largest group of citizens, you will meet on the square.
As soon as you are done eavesdropping, approach the exit guarded by soldiers and produce your ID.
A moment later, there will be an explosion, after which you need to turn around (pull the left analog stick down + X) and follow your companion (press and hold down the L2 button).
After you enter a nearby building, you will receive a first-aid kit from a woman, to tend to your wounds with. To do this, follow the instructions on the screen. This skill will come I handy many times throughout the game so, you'd better learn it. You also need to remember that, after you abort this action, your health bar will return to its previous state and you will have to start over.
As you follow Tess, you will reach a room with a cabinet blocking further passage. Move it by pressing the triangle and moving the left analog stick to the right.
After you jump down to a lower level, keep following the woman and press the X whenever you need to go around an obstacle.