Imagine the person pushing you is wearing socks and standing on a slippery floor while they push you. Their feet might slip out from under them and they would get pushed backwards at the same time you are getting pushed forwards. This is Newton's third law - if you push forward on something, you are actually pushing yourself backwards at the same time. We don't always notice this because we can brace ourselves against the ground, but the effect is always there.
So if you're in a chair and push forwards on the chair, you do push the chair in the forwards direction, but your hand gets pushed backwards at the same time. Your hand is connected to the rest of your body, which is connected to the chair, and in the end the whole thing is all pushed backwards just as much as it's pushed forwards. That's why you can't push yourself.
What you can do is sit in the chair and push on a wall. You push backwards on the wall and you get pushed forwards at the same time - Newton's third law. The wall is presumably braced against the floor and other walls, so you won't see it move, but you'll move forward.
Another way to say this is that in order to move, you need an interaction with an outside object. The wall is an outside object where you and the chair are concerned, so you can use it to push yourself forward. Another person is also an outside object. The floor is, too, so you can scoot along if you can interact with the floor. But if you don't have any outside objects, you can move the various pieces of your body around, but you won't be able to move the on average - something will always go backwards just as much as something else is going forward. Technically, we say that the center of mass stays in the same place (or if it was moving, keeps moving with the same velocity).