First off, the Q&A for falco's is "Dr. Peepee's Discussion thread". Ask anything there.
As for what to focus on, i suggest learning how to waveshine onto platforms, it's one of the main tools you'll use in continuing your combos. also learn the different ways to laser. Learn Isai drop lasers, ledge hop lasers.
also learn to properly shield pressure. this is very important. I'm not sure how advanced you are, but I find double shines and shinegrabs essential in falco shieldpressure.
essential? really? i never doubleshine anymore, and when i used to do so it didnt help me much. shinegrabs are a nice mixup but far, far from essential at any level and probably not worth doing until you're at a level where you have to use a lot of mixups to get any mileage out of your shield pressure because the payoff is very low compared to landing an aerial or shine.
i'm not saying there's anything
wrong with using them heavily, mind you - if that's what you prefer i'm sure you can make a totally valid style incorporating extensive doubleshine & shinegrab usage, but they're not essential in general.
shine wavelanding & laser tools, however, are actually essential!
however these are actually all technical answers. so now that i've spewed pedagogy all over this thread i will try to answer the question.
The short answer is: spacing, pressure, and reads, but you presumably already know that. For falco specific applications of these: Shield pressure, as Trahh mentioned, is important to have down. be comfortable landing your aerials at different heights (very early, very late, and middle at least).
learn how to use your gun right. you cant just shoot someone and win, and you cant just approaching shl twice and dair and win. you want to use your gun to:
-force your opponent in the air if he wants to stay grounded (marth, samus, others) or onto the ground if he wants to stay floating (puff, peach, others).
-clip opponents as they land from aerials, giving you time to pressure or grab or sometimes even just land an aerial.
-clip opponents if they're kinda out of your combo but still in a really bad position - a laser before they get to touch ground can mean the difference between another dash attack and a c-c-c-combo breaker
-force your opponent to shield when it's uncomfortable for them. if they can PS reliably this means you have to cut down on lasers a lot and dont use them when they're ready for it. takes a lot of learning. DD a lot. empty hop sometimes.
-not let your opponent be an ******* and wait for you to commit to unsafe approaches. do some lasering, get in their face but safely, dont get WD back grabbed
learn when not to use your gun:
-when you're running scared, you probably dont want to reverse laser. this will get you dash attacked or run up > shield / possible PS > ****ed a lot. laser if your opponent is giving you space to, but dont get fooled into thinking you have space/time to laser.
-if your opponent is moving freely and close to you you probably dont have time to shoot them. e.g. if fox is like 1 running sh away from you and you try to shoot him, if he chooses that time to approach you're ****ed.
bair and utilt are your friends. use them for defense.
ummmmmmm hopefully thats a good start?