Who do you play as? Avoiding grabs is very very difficult and smartly spaced grabs can become a real pain. What you need to do is develop a less linear playstyle by using platforms and jumps. The problem is being on a platform isn't so advantageous against Falcon as he can Uair you through it. Work on short hop aerials, pivot, and baits.
You can use R as @
nickthename
suggested, but another common technique is to learn to slide from down C to A, such that 1) you press C a short enough amount of time that you short hop and 2) you press A at the right timing for dair to come out early. This sliding method is useful for many things, like Fox's short hop double laser, Pikachu's short hop double aerials, etc. It's not always required to slide (you can short hop dair to uair with Luigi if you press fast enough), but it's faster than pressing.
Also you should always practice in the air and on an opponent (in training mode for example), because hitlag will mess up your timing for the uair and the z-cancel.