Zed has been a favourite champ for many since his release. He is a mobile assassin who can easily pick off and 1v1 champions with high dps, but lacks the team fight potential of many other champs, and also falls off late game. His early game isn't that good until he gets his main combo at level 3, and his ultimate at 6. He can easily be abused by many early game champions, but if he gets through laning phase, he can be deadly.

Pros
Fun to play
Lot's of outplay potential
Can 1v1 easily
Can pick off champions easily
Can roam easily

Cons
Falls off late game
Early game can be abused
Hard to learn and play
Useless if behind (no cc for team)
Gets countered easily (Stopwatch, Zhonya's, exhaust)

Farming is ideal for Zed because if he doesn't snowball early then his farm will make it so he will still do something for the team. His passive also makes it really easy to farm with because of his passive which periodically makes one of his auto's do more damage against enemies under 50% HP. If vsing a poke mage, farm with q from a safe distance and go for melee farm if safe. If vsing a melee champion, farming will be easy.

Laning phase:
Zed's laning phase isn't good for the first 2 levels because he can be easily outraded for extended fights by melee champions eg Yasuo. Wait for your level 3 and then you can use your main combo of W < E < Q procing electrocute(more in-depth in next chapter). If they engage on a fight when you're level 2 then you Q < E < AA which will proc electrocute and run away as you will be out-damaged for extended fights.

Against ranged champs in which you can't do melee fights, you poke them with your W < E < Q procing electrocute combo(more in-depth in next chapter), and slowly lower their HP. You also need to be careful of when you use this combo because if you use your w and the jungler ganks then you will have no escape and be forced to flash and maybe even die.

Post laning phase:
Zed's mid game is when he gets his power spike. When you get lethality items you can easily one shot enemy ADCs and squishies. What you want to do is avoid being seen and look for low health enemies and try to get picks. In the late game, when tanks start to buy armor, you will start falling off. When you team fight, your job is the assassinate the carries. You don't want to be the one to engage because you will be cced by the tanks and get killed by the carries. What you want to do is wait for the enemies to use all their cc on your team and then come out and assassinate carries.

Zeds main combo is his W < E < Q combo. This is his bread and butter combo and is what you'll be using most of the time to poke enemies in lane and in team fights. This is his main combo because getting hit by the shadow's E slows the enemies which gives you an easier time to hit your q's. If you land the whole combo then it'll also proc electrocute for a lot of damage. Most Zed's throw their shadows in front of the enemy which is bad because the angle makes it so that it is harder for both shurikens to land, therefore not procing electrocute. The best way to land both shurikens is to through the w beside the enemy or next to the enemy. This will give a better angle for hitting shurikens.

Another combo of zed is to W < W so you switch places with your shadow and then R < E < AA < Q an enemy and let the death mark kill them. This combo is good for picking off squishy enemies such as ADCs and Mages. Sometimes it doesn't work and the person you're trying to assassinate has a team mate that saves them. Well if this happens you just ulti back to your ulti shadow which will normally be far away from the battle since it is placed where you ulted and your enemy will be running away. Normally you escape but in some situations like if the escape shadow (R shadow) isn't far enough from the enemy then you will die.

Another combo of zed is the hidden shadow combo. This combo is probably the hardest to execute and actually find an appropriate use for. It is hard to explain so here is an in depth video on it by Redmercy.
Play Video

Another combo is one where you can kill someone and normally get out as well. You normally use this combo to outplay a 1v2 or something like that. This combo is most effective when uses over a wall because it essentially guarentees an exit, but is also effective when not used over a wall. So what you do first (if using on a wall) is you place your W over a wall and R onto the person you want to kill. When you get out of your R you instantly W < E which places you at a safe distance and your shadows do damage with your E. You then place your Q's to hit the death marked enemy which will then cause the death mark to kill the enemy. If you are doing it in lane then you do basically the same thing as above, except instead of putting the W over a wall you place it somewhere behind you or next to you (somewhere that is safe). This is effective for outplaying a 1v2 because when you switch spots with the W shadow, you can dodge some cc abilities that could've resulted in your death. If you can't understand this then here is a video which you will need to start at 2:41.
Play Video
There are many combos you can do with Zed but these are the most basic ones. The video above by Redmercy also shows you many other combos. There are also other youtubers who have posted combo videos so you can check them out on youtube. I also highly recommend watching ll Stylish as he is the number 1 Zed NA.