Welcome aboard. On the subject of Falco, he falls into the fastfaller category of combos, which essentially means there aren't any awesome Training-Mode style combos on him until about 50%.
Generally, the accepted practice is to use N-airs to keep him at bay and space while dealing chip damage up to mid percents, and then you're going for a launcher -> knee combo.
Vertical launchers work best, which means side-B and stomps. Side-B is risky, but it's your best launcher because it launches them the highest at low percents. It can also link to itself (don't get carried away), but generally I grab to build percent. You should be all over him with stomps (land behind him so you don't get grabbed, and get good at moving after you whiff one), however, because if you land one, that generally means a free grab, knee, smash, side-B, or if you are feeling frisky another stomp (basically whatever you want).
At about 50% or so, you should be gunning for a quick combo into knee to get him off the level, following that, edgeguard. I like to use a down-angled F-tilt at the ledge to knock away a potential illusion, then either repeat that, edgehog or stomp his firebird.
If you can get him to shield, go for a grab. Use your dashdance, stomps, and neutral airs to accomplish this; generally he'll have gotten himself in a bad spot if he's sheilding, and will try to wait you out. If he dodges, then you're generally not too bad off, but if you get one...
Up-throw is golden! At around 80-90%, any DI but away, this leads to a forward smash (see G-Reg Combos 101). At lower percents, chase his tech with a stomp, knee, or regrab. If he doesn't DI or tech the upthrow, then you can upsmash him and knee him from there at just about any reasonable percent (under about 70 or 80). He'll learn quick though, so try to practice your tech-chasing.
Another added advantage of going for grabs is that he'll start to stop shielding, or otherwise noticably modify his behavior, which you can take advantage of with more stomp and knees and stuff.
Anyway, recap: Low percents, build damage and go for grabs. Mid percents, look for a knee combo with your launchers. High percents (90+), everything starts to really combo, and the higher you go the better; u-throw to F-smash, and N-air to knee and most of the combos you'll find in the combo section of AngeloBangelo's epic guide to weightlifting Falcon.
Did I miss anything? Ooh 600 posts.
( I'll post vids soon)