Azir is a pretty complicated champion. He has especially low defensive stats, so he is pretty unforgiving if you aren't trading or positioned properly.
The trick with him is playing the range and CD game. You abilities have good range, but you still have to get within range of the enemy's poke to use them. Try and wait for/dodge the enemies abilities, then go in with your combo. W - Auto - Wait - Q - Auto if you can. Try placing your W just in range to get an auto off before following with a Q - Auto again. But be mindful of taking harass back as you basically root yourself when you cast Q and auto.
If you are new to the champ, just focus on this. Long-range poke for lane-presence and just farm safely. Don't try to all-in even if they are low unless they make a big mistake, because Azir's burst is mediocre. Once you get better at farming and poking, then you can start learning how to abuse the utility in your E and Ult.
Teamfights are similar. Try to not eat enemy abilities while casting your own. Dodge the abilities then walk up and spam W - Q and auto as much as possible. Stand in the back and ult away any enemies that try to dive you or your back line. When you get positioning down, you can position yourself such that you are basically untouchable. Always try to summon soldiers in auto-range so you can do some damage before you need to Q to re-position them.
Azir does so much base damage and has such good scaling right now that you can do well with him without even really worrying about using your ult and E offensively. However, once you master him and start using these in roams and skirmishers, you will absolutely wreck teams.
For Item build and skill order I recommend this:
Max R -> Max Q -> Max W -> Max E (once you get better, I suggest E max second because that gives you better chase, and all-ins. W Max is better for staying safe and poking).
Build Doran's Ring -> Morellonimicon -> Stinger (don't upgrade. If you have 10% CDR from runes and masteries that gets you to 40% CDR really fast -> Sorc Shoes -> DCap, Void, Zhonyas (whatever order you need).
Sell Stinger in the really late game as its only really used as a cheap mid-game powerspike.
Some people may argue against building stinger, but that's how Faker builds Azir, and I think Faker probably knows what he is doing ;) I also find the extra Attack-Speed let's you get A LOT more autos in, which is great. I prefer it myself.