Falcon Punch is all you need.
JK. I use nair and bair in neutral a lot. But the most used moves are d-throw/f-throw and uair with knee. Don't understimate dair, it's fast and lethal, and also comboes with knee if the enemies bounces in the stage.
Tilts are good, but I don't really see people using d-tilt that much. But if you hit dtilt, it sends them horizontally, perfect for some offstage magic, or even an opportunity to use the heel of humilliation (utilt).
Basically, almost every move of Falcon is really good. There's no move you should discard because of its lack of effectivity.
But remember.
The hype button is the . Just be careful, don't use more than twice the side B, and don't spam falcon kick. Use it to punish rollers, but also use your dash grab.