Those are some rather open-ended questions.
Lets see...
If the opponent is coming down back to the stage (eg after a dtilt hit him at high percent but didn't KO) and you believe you can rush or out prioritize their defense (dair, usually) and could punish an airdodge. Depending on which way it knocks them I usually follow this up with another uair (especially if I still have my second jump), a fair, a bair, or a zair.
The uair works especially well on opponents who are above you on platforms. You may even be able to uair upwards into them and again on the way back down.
If your opponent is directly below you the uair can be used like Fox's dair to hold your opponent in place. Once you land immediately attack to follow up on it. I usually either dsmash, dtilt, jump and uair again or screw attack. Screw attack works great on stages like battlefield if you're under the middle platform - you may land in time to follow it up even further against your opponent.
In general to string a number of attacks against an opponent you want to hit them in such a manner that they'll lag long enough for you to hit them again (so your move can't be to laggy) but not get knocked too far away. Good moves for these things are the uair, fair, dash attack (into screw attack), jab, canceled missiles, charge shot (especially at full charge where they just float in the air before getting knocked away) possibly the dtilt or dsmash, often the using the dair to po 'em up and of course the zair. End with a move that knocks your opponent far enough away that they can't just start a combo back. These moves include smashes and the bair.
Its best to just watch videos, honestly, to get a better idea, and then to try them out to get a feel for them.