To the people who recommend that Swil simply use his ground moves to push an opponent into the air, please realize that that is the MAIN PROBLEM with Falcon. His priority lies in the space above him, and he has no consistent method of putting the opponent there. Approach styles vary from matchup to matchup due to their limitedness, thus making combo starters (well, aerial followup starters in this game for lack of combos) even more difficult to manage. All of his options are extremely situational and, against a campy projectile character like Mario, are swiftly put to shame and uselessness. Please, let me know if you can come up with a universal or even relatively reliable ground approach with Falcon, then I might decide to main him again. In the meantime, realize that this is his most serious handicap, and brushing it off simply by saying, "use ground attacks to initiate aerial combat," is overlooking Falcon's greatest detriment. If anything, that deserves the most strategic discussion of all of the things mentioned in this thread. It's a crime to just jump over it like that.
That said, and I don't mean to completely slam the poster above for glancing over the handicap that I just mentioned, let the ground game be an understood and inherent problem in Falcon's style. If you have any specific insights, offer them, cause I sure as hell don't have any and I know the rest of us could use them. But don't just postulate that ground->air comboing can be done easily and give advice based off of that, because that's really not progressing our tactics at all. You can't just skip over the first roadblock and make everything else happen. Let's start from the ground up here (pun not really intended, but duly noted).
And mind, using the Falcon Kick on characters with projectiles is somewhat risky, as you lose a lot of the momentum that would carry you past and space you properly for followups (that is, momentum lost in colliding with projectiles). I like using the Falcon Kick in Brawl, too, and even more so than the Raptor Boost, but both are to be reserved generally for surprise/mix-up approaches. Try to use Falcon Kick when the opponent is at higher percents and Raptor Boost at lower ones, though against ranged opponents I suppose the Kick is really the only viable approach of these two, regardless of current weight.
Oh, and my issue with using the infinite part of the jab combo is that, if let run for too long, you get trapped in this range where they've DI'd out so the hit stun is gone and they're in range to smash you. The fists have incredibly low range for how deep they put you into dangerous territory. I admittedly use the opening part of the infinite sometimes for damage, too, since even doing a jab->throw combo is not a sure-thing if your opponent is on his toes, and I really need to get that extra damage. I tend to not let it get past the very first part of the infinite though. That is guaranteed trouble, no matter how you look at it.
Oh, and as for the few tips on ground game I have, here are some situations I've drafted:
For tall, slower characters, shield grabbing is effective. Shorter ones like Kirby make this tricky, especially on slopes. Also, falling nair->throw or nair->insert followup here chains are good if spaced properly.
For shorter characters, just suicide. This is why I gave up maining Falcon. I try dairs, shieldgrabs, dash attacks, Falcon Kicks, and Raptor Boosters here where I can, but against people like Olimar, the ground matchup here is dismally difficult. I have no advice, just take advantage of any opening you can with any of these moves I mentioned. Judging by your style you know this already though.
I suppose you could technically do a falling RAR or land behind them with a bair too, that is also another workaround I sometimes use. None of these is really that reliable, though.
Dash attack against fast characters that don't expect it. Dash->Utilt sometimes works, Dtilt->spotdodge->throw also works sometimes, somewhat universally (it depends on jumping capabilities, but this generally works. They usually retaliate quickly after the dtilt too, so that's why I spotdodge or at least avoid after the tilt. Your choice).
And that's all I have to offer for tonight. It's sort of convoluted and really lacks any direction, but glean from it what you can. I think I was just rambling for the past 30 minutes, in all honesty.