Keep going to reach a room with five pistons. Some of them go up to the ceiling, so don't stand on them for too long or you will be crushed. The middle one, however, does not go all the way up to the ceiling, so stand on it and when you see the ledge way up near the ceiling, shoot a portal onto the wall back there and drop down. Shoot a portal here of the other color and go through.
Moving on, you eventually get to another area with some pistons. Shoot a portal onto the low ceiling where you can see a horizontal piston, and then shoot a portal onto the even lower ceiling that a piston is smashing against. Stand on the piston under the portal and you will drop through to the area above.
You should land on a horizontal piston (but if you somehow don't, just run back to the portal that you came through and try again). Jump across to the other horizontal piston when it goes across to reach a ledge.
Shoot a portal onto the high part of the wall here, and use floor portals to get to the ledge near the ceiling. Then in this next area you will see several high-speed pistons moving around dangerously. Shoot a portal beyond them (it might take several tries), then shoot a portal where you are so that you can get through.
To your side you'll see a couple of vertical pistons. When they both move down, shoot a portal on the wall beyond them (you don't have a lot of time to do this). Then make a portal where you are to go through.
In this area, there are a bunch of horizontal pistons above you. Shoot a portal above one of the pistons when it retracts, then hop through a portal to get on top of it. You can stand on it even if it's all the way retracted. Next, make a portal on the opposite wall, above a piston, and use a portal to get there. Finally, you should be able to make a portal on the opposite wall that will drop you onto a sideways horizontal piston. From that piston, shoot a portal onto the floor of the ledge opposite. Use a portal to get there.
Drop down where you see the cube tube. You can walk along the tubes for a while, and you eventually reach an area where a horizontal piston is right in front of you. Shoot a portal onto the wall above where a horizontal piston is hitting the wall, and then make a portal of the other color in the area where you are, and stand at the portal until the piston extends to where you are. Jump onto it to get across to the ledge.
You will reach a door that doesn't open and hole in the floor. Drop down into the hole.