Falcon's best moves to use when approaching or in neutral are jab, grab, SHFF nair, SHFF bair, SHFF uair, ftilt, and dash attack.
Single jab is especially great because it's so fast and has such little ending lag. You can catch rolls, spotdodges, and the start up of laggy attacks with it and then confirm into a gentleman followed up with all sorts of stuff. You can also use it to poke people's shields and then grab them.
Also another tip is you don't have to go in all the time with run up grabs or dash attacks. Sometimes it's best to wait at the other side of the stage and see what your opponent does. Try to figure out what their attack plan is and then counter it.
If they try spamming projectiles, let them spam for a bit until they go on autopilot, then catch them off guard with a run up SHFF nair into grab.
If they like to roll behind you whenever you rush at them, instead run up, but then stop when you are at around roll range and see what your opponent does. If they roll behind you at that moment, you are ready to punish with a grab or jab combo.