This matchup can be more difficult than it should be if you don't know how to react to falco. The main thing to watch out for is the pillaring. If you get caught in one of his pillar chains, expect a solid amount of damage racked up on you, despite your attempts to DI. The most you can do in this case is to tech dair ASAP and make sure your roll isn't readable. The most efficient way to reduce the likelihood of you getting pillared is to stay in your shield when he's pressuring you. Attempting to shield grab after the dair hits will just make you eat a shine, and into the pillar you go. Just shield until Falco, usually instinctively, hops out of shine, and back away.
Tech chasing is important. I personally like using dthrow, but mixing up uthrows at low percents can work too. When the falco starts to catch-on to your tech chases, and either techs in place or stays down, use a knee or stomp to punish. Proficient tech chasing is going to give you control of the match and is crucial to netting that win.
As for this whole chain grabbing business, you can do some uthrow chain grabbing around 70ish to late 80ish percents. Continuity of the CG depends on their DI though. Still useful, and if they do DI out of reach, chances are they will tech and you just read from there.
Uthrow to fsmash around 90 percent, and anything higher towards 100 can be uthrow, nair, knee or stomp, nair, knee.
Spacies are probably my favourite to fight, despite how difficult they can be. Good luck.