So I'm not a pro but I've played Link a lot through each of his incarnations of Smash, so take this with a grain of salt.
And anyone, please correct me or share stories/theories in response because I'd hate to be misleading by mistake.
On aerieals:
I have problems linking the two hits of Fair and Bair together; FF Nairs into jab are my aerial weapon of choice. Any tips there?
This is one I picked up during some brawl time and it translates over to Wii U well enough for me. Short hop for you Bair and Fair. Remember, you can actually commit your Bair and Nair at the exact moment you jump to, so as you're rising you can keep hitting (best I've ever done is 6 hits with Bair: 2 on first hop, 2 on second hop, 2 on fall).
ATs:
What ATs do you recommend first learning?
I'd say Short Hop and Fast Fall. They're just big time savers. After that, I'd learn a little about bomb sliding and S-Smashing. (Can someone remind me if we have DI back? I'm pretty we do. If yes, then learn Directional Influence).
Zair:
What's the best usage of Zair? Poke with it into a jab on landing?
Zair is a beautiful thing. It's good for spacing and for a surprising offensive attack when your opponent thinks you're going to dive in with a Fair or Nair. Additionally, you can be really creative with comboing off Zair. See, if you pull a bomb, then hit Z while you're midair, you'll throw the bomb. But if you pull a bomb, air dodge, then quickly hit Z, you'll Zair, then you can throw the bomb. It's a good little surprise when they think they know what's coming.
On a different note, if you leave the Stick in the neutral position and hit Zair while holding a bomb, you'll drop the bomb directly below you. Every so often, someone panics about that and it's hilarious haha.
Main rule of thumb, don't be predictable. Link is a really fun character because he has so much versatility and utility - be aggressive, defensive, adaptable, and whatever else you need. Hope that helps some!