Puff's aerials can come out very quickly after a jump, that's probably what you're attributing to air speed.
You can input a jump and then toss out an aerial nearly instantly then do it again very quickly.
A thing to practice if you're very new to puff is staying airborne. Puff's SHFFL is not that useful, and it is not wise to use in most situations. Stay low and airborne, land in autocancel timing from nair or with no aerial out. Puff is not like other characters who need to land after shorthops to prevent vulnerability, you have extra jumps to exploit and fast, effective hitboxes to use.
That being said, Puff is probably the hardest viable character to play at a high level, so hard that there's only one top level puff player (Hbox), good luck.