Yo, I main doc and I secondary falco, so I think I can give you some useful advice.
First and foremost, you can approach with SHB, but it wont be your bread'n'butter in this matchup like it is in most. Granted Doc can obviously cape it (which is your main problem, since it'll completely throw off your timing), Doc wont always be ready to cape it, because if he is, he cant really be ready for anything else.
As for pills, dont forget that your shine is actually a reflector (lol). Dont make this a habit, but if doc tries a pill rush, you may want to try to a JC shine (or whatever it's called, shine the pill and jump out of your shine). That way, the pill will most likely go back to doc and you have a few options out of your shine.
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Obviously, you dont want to get grabbed. Doc can chaingrab falco pretty far, so try to stay away from his grab. If you do get grabbed, get to work with the DI.
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As for combos, you have to avoid his combos just like he has to avoid yours. You have to watch for for u.air to d.smash combos (and this may be one of his approach options). You also need to watch out for multibutts (u.smashes) from a grab, or any missed tech you do.
The only other "common" attack you really have to keep an eye out for is the b.air. Doc's b.air is ridiculously fast, and although it doesnt have a great knockback, it's enough to get you moving, which might get you off the stage.
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If and when you recover, you have to be very careful because of the obvious reason, the cape.
Dont be scared of the cape, though, especially if you can make it close to the ledge. The closer you get the more options you have.
On average, most doc players arent ready for forward + b unless you telegraph it (i.e., you ALWAYS do it to recover), so if you you may want to use it to dart back to the edge. Also, you can still get away with ledgehop -> forward + b, but again, you CANNOT train doc for it, or else he'll punish you for it sooner or later.
Of course, if you ledgehop - double laser, he might have the cape ready anyway, so you have to be careful about forward +b.
But anywho, the worst position you can be in when recovering, other than not being able to make it back, is being under the ledge. That's just asking for doc to cape you.
To avoid this, try up + b'ing up (just like you'd avoid anybody who's standing on the edge to edgeguard you). If you get him to miss his cape, then you can fastfall to the ledge and be aight.
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If you want some stages to take doc to, I personally say anything with platforms works.
Platforms mess up docs combos, and gives you a chance to escape chainthrows and multibutts while extending your combos.
FD and PS also work, but you have to know that you're giving doc open lanes to pill rush. PS isnt as bad since neutral, wind, and water have platforms for you to jump on (and again, to use for your combos), while fire and ground have walls to stop docs pills.
From my understanding, Falco works pretty well on both moving stages (correct me if i'm wrong). Doc on the other hand, isnt really spectacular at them, and actually loses alot of pill rushing options since there arent many open flat areas. So if you can handle playing either moving stage, that might be a good option against doc.
The one stage you probably want to avoid though is corneria. Doc can get some weird pill trajectories here, and he can also use the fin of the ship to up + b to get high-priority hits and keep you pinned near the door.
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All in all, doc is a "different" fight, imo, since you have to watch for the cape, but it's not an almighty cape. Just be careful and be tricky when you recover, and dont rely too much on SHB for approaches (unless Doc isnt caping, then by all means ***** it to the end of the earth) and you should be fine.