This page of The Last of Us game guide contains a walkthrough of The Museum stage as part of The Outskirts chapter. During the exploration of the title building, you will be forced to deal with the new infected - by either sneaking past them or killing them. During the march, the heroes will come across a room with a table used for upgrading weapons.
After you get out into the streets, turn right and you will reach a truck standing across the way.
To make it over to the other side, you need to move the container towards the truck. You can fid it in the building to the left. Before you enter, start listening out and turn your flashlight off. Inside the building there are three zombies waiting for you. The creatures do not move so, you can easily sneak up to them from behind and strangle them. First, deal with the beast to the left, then the one to the right. This way, you will be able to squeeze past the last one safely.
Once the area is secure, search through this location and move the container towards the truck.
After you walk over to the other side, open the gate to the left and dash inside to avoid the approaching zombie.
Inside, you will find a workbench, where you can upgrade your weapon with the spares that you obtained earlier. It is a good idea to increase the clip capacity and the increase reload speed to your favorite weapon.
Once you are done, walk over to the nearby door and go left. This way, you will reach a location with collapsed floor.
Before you climb up, though, explore all of the areas because, there is some useful gear stashed here.
Once you limb up, you need to find a passage in the room to the right.
Press the triangle to enable Tess and Ellie to make it to the other side. Still, you won't be able to do it yourself. After a short cutscene, turn back and go left. This way, you will get to a corridor with clickers walking around.
To avoid getting into fight, turn left and follow the first enemy you encounter, until you will be able to take a turn to the left. In the room, you will find a door that you need to open with your blade. Craft the appropriate item and explore the area behind the door thoroughly.
Once you are done, walk over to the location at the other side of the corridor, while minding the creatures walking around here. You now need to turn left and follow the zombie there.
Ultimately, you need to get to the other end of the corridor that you have just been in. You can get around the enemy, patrolling this area, only through this location. When the enemy starts going in the opposite direction, turn right and open the nearby door.
Past the door, there is a staircase that leads up.
Once you climb up the stairs, turn your flashlight off and eliminate, silently, the enemy banging at the door. You will then be able to walk through the door, where there is Tess waiting for you.
Once the woman eliminates one of the assailants, quickly move to the next room. There, you will have to face a swarm of runners. To eliminate them quickly, it is best to retreat to the smaller room and shoot at the enemies one after another. In a situation when one of them approaches too much, punch him several times.
After the last one of the zombies dies, approach the broken window in the corner of the room.
You need to talk to Tess there and then, climb up the fire escape to the rooftop.
Once you get there, find a plank and pick it up by pressing the triangle.
This way you make a makeshift bridge connecting this building with the other one.