Head for the gate right in front of you and pry it open with your machete. Exit through that gate, then turn right and hop the fence.
Exit the alley and turn right once you're back in the main street. Visibility is noticeably getting lower at this point. Just keep going down the street.
There are some cars that will block your way but the street is generally a straight path. Just open your map every now and then to see where you're headed. Head inside the mall once you're there.
Once inside the mall, follow the linear path and turn right at the end. You'll find an escalator there, use it to get to the next floor.
Go around the left side of the escalator and you'll find a shop with its doors open. You'll notice that the back wall has some climbing posts. Jump on it and make your way to the next floor using those.
You'll be using up a lot of your stamina just making it through the first half of the climb. Use the piton that you found lying around the floor to create a rest spot for yourself before you continue up the next floor.
Leave the shop and turn right. Climb up the collapsed part of the building to get to the next floor then enter the plastic protected shop on the right side and keep walking until a cutscene starts.
Grab any supply that you can find in the room, then head out and drop down on the elevator shaft.
When you get to the lower floor, move over to the next elevator shaft and make your way up the railings inside it.
Climb up through the open elevator door a couple of floors higher, then turn right and head across the bridge.
At the end of the connecting bridge, look to your right and you'll see some exposed parts of the mall structure. Climb up the next floor using these.
Make your way up the ledge and press space to climb up on it. Some ledge can be stepped on so always check for that when you're climbing. Saves you the hassle of having to use items.
Shimmy over to the left and drop down when you reach the billboard, then continue over to the left side while hanging on the ledge. Climb up when you finally reach the other end.
Go to the edge of the building on the left side and you'll find the telescope. It would be glowing in white so you can find it quickly.
Use the telescope then go to the right side of it. You'll find a makeshift elevator. Grab the jerry can and put the gas inside the generator, then bring the elevator down.