There are small, immobile turrets in this room that will shoot at you if they see you. To defeat these turrets, all you really need to do is make them fall over. This can be done by making them fall through a portal, or dropping something on them and making them tip over. You can even pick them up from behind, lift them, and drop them, and they might tip over.
However, when they fall, they shoot a bunch of bullets for a few seconds before deactivating, so you should keep your distance when making one fall over.
When a turret sees you, it will shoot at you with a rapid fire weapon until it loses sight of you. Getting shot won't kill you instantly, so if you do get shot but manage to get to safety quickly, you'll recover your health and survive. But if you get shot a bunch of times in a short amount of time, then you will die. So try to avoid that.
As you enter the test chamber, there will be a turret with its back to you. You can walk over and pick it up, but the easiest way to get rid of it is to make a portal under it and then a portal of the other color on a wall or the ceiling. It will shoot like crazy and then shut off.
Proceed through the chamber and you'll see a turret's laser hitting the wall. Carefully walk toward the laser, making sure not to walk into the beam, and look through the door until you see the far wall. Make a portal there, then another portal of the other color in this room, away from the laser. Go through to end up behind that turret. To be on the safe side, disable it by shooting a portal below it and shooting another portal on a wall or ceiling.
Continue on and there's a laser shining at the wall. If you're not trying to get rid of all of the cameras for the Camera Shy achievement, you can just make a portal on a wall in the distance to get past this turret. But if you want to get rid of the cameras, carefully move forward until you see the turret, but move back quickly or it will see you and shoot at you. Make a portal on the wall behind that turret, then another portal on a wall nearby, and go through to disable the turret in the usual way.
Next, you will see a piece of wall extending out, with two storage cubes behind it, keeping it open. Move those cubes aside (you'll need them soon) and then you can go into the small room through a crack in the mesh behind the moving piece of wall. I'm sure you've been suspicious of the goings on here at Aperture Science, and now with the word "help" written in blood and with this little, dingy room, you now see clear evidence that this is no normal testing facility. Indeed, something seems very wrong here. Someone had to hide out in this room, with nothing to eat but a few cans of food. Be sure to read the walls....
Anyway, exit that little room, then carefully move such that you can see the ceiling above the turret that sits behind the glass wall. Make a portal on the X on the ceiling (it's OK if you get shot a little, as long as you can move to safety), then make a portal on a wall somewhere safe and put one of the cubes into it. If you positioned the ceiling portal correctly, the cube will smash the turret and disable it.
Next, carefully glance down the hallway where the other laser is shining from. There is another X on the ceiling down there, so you'll have to shoot a portal onto it quickly so the turret down there won't shoot you. Once the portal is there, make a portal and put the other cube through to disable this turret.
Next up, there is a turret that will see you if you try to go over to the turret that you disabled at the end of the hallway, so be careful. You can move down that hallway until you can shoot a part of the ceiling above that turret, then drop a cube or a disabled turret onto it.
Once you have disabled that one, there is another one farther away in that room. It should be easy to shoot a portal above it, though, and you can probably even just sneak up behind it and disable it by tipping it over, since it's aimed away from you.
Now, be careful looking through the far door of this large room. You can see a test chamber lock in there, as well as a button, and of course a turret. It's not too hard to shoot a portal above that turret and drop something onto it, so do that.
There are two other turrets guarding the big red button, and one of them is facing away from the door into that area, so sneak up behind it and tip it over.
There is one final one that is aiming toward the test chamber lock. Move such that you can shoot a portal onto the X on the ceiling above that turret, then dump something onto it.
Now you can drop something onto the red button to go through the chamber lock, but don't go through right away. There's a turret aiming directly at it on the other side. This turret is behind some mesh, but the mesh doesn't block shots from the portal gun, so just make a portal behind the turret in there, and go through to kick over the turret and disable it.
From the mesh room, you can see another room, with a turret facing away from you. Try shooting a portal onto the ceiling of that room but as close to your room as possible. If you hop down through it, you should end up behind that turret and you can just tip it over.
From there, you can go into the elevator to the next level.