Now I know you just want to jump right in and start screaming, but unfortunately that isn't going to happen. First you need to know how to properly use your diaphragm and properly support your scream with it. It's the muscle that basically allows us to breathe. There are a couple ways you can "activate" it.
Method 1: Act as if you were taking a massive shit. Yes, that's right. Pretend to take a massive shit and push it out. Now hopefully you didn't shit yourself in the process. Did you feel your muscles contract/flex? Most likely that was your diaphragm doing all that work.
Method 2: Just give a quick shout. As if you were shouting at someone across the street. Just a quick "Hey, Bitch". Did you feel your muscles contract? If so, then it was most likely your diaphragm again. See, isn't using your diaphragm easy? If you still aren't sure if you are using your diaphragm then watch the video in the right hand side bar. Now before you move on, you must engrave this into your brain. YOUR DIAPHRAGM IS THE MOST IMPORTANT ASPECT IN SCREAMING. Take a moment to get that stuck in your brain, please and thank you.
Alright, so you want to learn how to False Chord scream. To me it was the easier of the two screams to learn. There are two ways I have figured out how to achieve the base to this scream.
Method 1: Sigh roughly. Lets say your parents tell you "Little Billy! Can you please go take out the trash!?." Obviously you don't want to take out the trash so you sigh pretty hard. <Insert rough sigh here> Did you feel some a slight gurgle? Or maybe your sigh became sort of raspy. Now take that gurgly/rapsy-ness and use your diaphragm to PUSH more air out. Don't use your throat to force the air out, use your DIAPHRAGM! Now hopefully you did it correctly and it didn't hurt. Now if it hurts, stop for a minute then try again. If you can't understand the concepts I have written, just go ahead and check the video.
Method 2: Bark as if you were a dog. A deep "ruff" will do. Now simply do that and push with your diaphragm until you get a sound that resembles a scream and just mold it from there. Again, if you don't understand; watch the video.
Fry screaming does not come naturally to most, but it is easier on the throat and can be held longer than an False Chord scream. Here are a couple methods I figured out works best for MOST people.
Method 1: Hum, and slow it down til you get the noise that the grudge makes. Now if you don't know what that is then just google a clip of the movie. Now once you have that noise, put a pencil/pen between your teeth and try to move that noise ABOVE the the pencil/pen. At some point it will start to get kind of "weezy" or "screechy" and thats a good sign. So once you get that, start pushing a little more with your diaphragm. This is a very delicate process. You have to figure out the exact amount of air needed to produce the sound. It just comes with practice, so don't give up if you don't get it right away! Again, if it hurts, then just wait a few minutes and try it again. If it continues to hurt then come back, read this again and give it another try.
Method 2: Take the grudge noise and then blow out some air as if you were trying to fog up a window. So do the grudge noise and "fog up the window" at the same time and see where that takes you.
Method 3: Hum and then just make a squeaky noise at the same time. Now I didn't figure this method out until I had already learned how to fry, but I found it to be one of the easiest to understand and do.
Method 4: Do a falsetto (again if you don't know what that is just look it up) and then take the grudge noise and work it into your falsetto and they will sort of "balance" each other out. Add the grudge noise and it's going to bring your falsetto down to the point where your voice cracks, which is exact where the fry comes from. So give that a try and see if it works.
How To Use Your Diaphragm
(Yes, this video deals with singing, but the concept is exactly the same with screaming)
These are the videos I feel do the best job explaining HOW to do the fry scream. So watch them if you need to.