Zed's laning phase is definetly one of the strongest laning phases in the game. Your dueling potential is very strong, but you need to land your Q's. You have several different ways to lane:

Defensive
Farm with Q and W - E. Wait till 6 and you can maybe go in, depending on the matchup.

Aggressive
Spam your full combo whenever it's up, and also try to get behind your opponent with W. This will make them feel umcomfortable and unsecure in lane, which is good for you.

Poking
Use W - Q whenever you have the right opportunity.

JUnglers usually doesn't like to gank Zed, and if they're below fifty percent hp you might actually wanna go for the kill at level three plus. If you can land the full combo with thunderlords, you can basically kill the enemy jungler while maybe burning you flash to get away. If you're getting camped make sure to save your W and farm with Q's as much as possible.

When to kill as Zed
First of all, if they have both summoners you probably wanna wait with your ultimate until you poke them below fifty percent. If they have something like barrier or exhaust, you wanna be very picky with when you wanna go in with your ultimate. If you fail your ult, it has a long cooldown so you don't want that happening.

This is where Zed truly shines. In teamfights you also have several different options as to what you want to do.

Assassin
Deathmark a backliner, take them out and go back to your shadow. This is usually the one you want to go.

Spray
If you have a team with alot of damage you might wanna do this. W in Q and E as many enemies as possible and you have a big advantage with your damage. If you're going spray, you should get last whisper early.

Splitpush
If your team is starting to lose a fight, go down there and clean the enemies up with your ult and general output damage.

Late game Zed is exactly the same as mid game

Zed has really strong roam. You usually wanna go for the enemies adc rather than the enemy's top laner.

When to roam
When you have pushed up your lane, it's a perfect time to roam If you go for bot, there's a few champions to watch out for.

Bot lane counters
Braum, Leona and Thresh can greatly counter your roam. You don't want to roam and fail it, that is the worst outcome. This is obviously still a possible roam, but generally you wanna pick a very specific time to roam vs those such as when they're below fifty percent health.

Good times to roam top
If you wanna roam top, you should usually do it if your team is behind in gold. Roam, kill the enemy top laner, push the tower. If you suceed it will help your team greatly.