ZSS has changed a LOT since Brawl, so the skills that worked in Brawl will not transfer over too well.
Her Zair is like most Zairs. Hitting it will take practice and is more useful against some characters than others. Characters like Jigglypuff will not be good characters to use it on. Bigger characters like Bowser will be good targets. The easiest time to use it is against characters that frequently use short hopped aerials. If you really want to know about Zairs though, you might want to poke around the Samus boards. They'd have more info.
Following up with throws gives you a few options. At mid/high percentages, Fair is the way to go. This will generally put them off the stage, and put them in an awkward positions.
At lower percentages, see if you can run forward just a bit, and then go for an up air. Up air is easier to follow up on (with another up air most likely). You can also choose to not follow up directly. You can wait on the ground, try to bait an airdodge and punish with Up B if they're at KO percentages (120 or so).
Basically, you have tons of options, so mix it up depending on the character and situation.
Again I'll repeat, ZSS is an entirely different character in this game. Her spacing game has been nerfed hard, but she has far better mobility options. She has some really interesting options that will be tough to deal with.
