First off ill start off with introducing myself, I'm ceshh a silver Zilean support main (impressive I know). I have over 120,000 points on him and M7.

Now I really haven't been playing league for that long. Actually about 6 months.
Which obviously to other Zilean mains, or support mains in general (that have been playing longer than me) seems ridiculous for me to make a build guide so early into playing the game and maining a champion. But for other Zilean mains that have been playing the game for a while or to new comers to the champion. I highly recommend trying out this build a few times to see how it works for you as changes with the meta have lately altered the play style of champions like Zilean.

Zilean is imo an underrated support.
His passive allows you to level up your teammates when they're close enough as you gain experience over time.

His Q throws a bomb which will prioritize enemy champions in its range, otherwise it will go onto a minion or just the ground. However if an enemy champion or minion walk across it while it's on the ground, it will be picked up. The bomb will explode after around 3-4 seconds after being thrown.

His W lowers the cooldown of his Q and E by 10 seconds. This allows Zilean to throw two bombs at the same time. (Q, W, Q) When 2 bombs collide onto a target, it stuns within its range, exploding once initially to stun and then a seconds time shortly after. (If the initial stun kills the target the two bombs will blow up simultaneously.

His E is his greatest ability. Not only can you slow enemy champions down but you can speed up allies by up to 99% at max level.

His Ultimate (R), if put onto himself or an allied champion before death, will resurrect them.

Sorcery

Arcane Comet - Is great for a little extra poke damage in laning phase, pairs really well with the double bomb stun.

Manaflow Band - Mana is crucial for Zilean, getting mana back quick is extremely useful especially early game. Procing Manaflow Band will help you with sustain.

Transcendence - CDR is crucial for Ultimate CDR. (And W, so speeding up is faster and stuns are more frequent)

Scorch - More poke
(Water walking is an optional last pick as well if you want to roam frequently)

Inspiration

Biscuit Delivery - I get these for sustain in laning phase

Cosmic Insight - Extra CDR for Abilities and Zhonyas/Twin Shadows

The Skill Sequence is simple.

First ability will be Q to help that slight bit with leashing your JG.
Your Q is also a very good early game poke as they won't have any early game magic resist.
This will be the last ability you max out.

Next get W to throw in some early game stuns. Don't go for a stun unless you know you can get it. Your mana will deplete fast. I usually save stuns for saving a teammate if they have a support that can pull them in or any other CC.
This will be the second ability you max out

Next will be E which you will nearly max out first. Rushing that 80% speed/slow is crucial to stopping enemies and saving teammates. Trust me your ADC and JG and whoever else roams will love you for having their backs.

Now obviously get your R up whenever you can. Your ultimate can take some really fast reflexes depending on the situation. Or, if you need a teammate to tower dive to get that extra kill it comes in very handy.

Flash - There's not much that needs to be explained about flash. A quick getaway spell that everyone should use all the time.

Ignite or Exhaust - Whichever one you take is gonna depend upon your ADC's play style. In Solo Queue I take ignite by default unless they suggest exhaust or heal, however, heal is a very self centered item and is better for your ADC to take it.

Overall Zilean is a ridiculously good and underrated support if you couldn't tell.
He's probably one of the hardest supports to counter unless you're a Yasuo or Morgana.
Your fast cooldowns allow for easy assists and your CC is insane.
Not only that but you can save your teammates from death, and get them slightly ahead with your passive.

Now, when it comes to playing in game there's some notes you need to take.
Stay. Focused.

-Always pay attention to your teammates mana and health.
-Use your E to help teammates get away from death, and take the death yourself if you have to. It's better to let the support die than any other laner (unless they're just inting and completely useless).
-Don't use stuns too much early game. Your mana will suffer.
-Use single bombs early game to get in good poke.
-Use your ult on anyone you can that is about to die (unless they're feeding, at that point just let them die if you know you can use it on another teammate very soon).
Your ult during team fights can take a lot of focus to scout out who needs to be saved and who doesn't. Now while paying attention to that you also have to make sure that you aren't getting obliterated by the enemies as well.

Combos

My favorite combo is; E, Q, W, Q, E. Very basic, Slow the enemy so they can't escape, stun them so they're stuck, slow them again so they really can't escape. By this time your teammate should of cleaned them up (obviously depends on the situation).

Anything else I do is just single abilities used to poke.
-Single bombs to get in a small amount of damage.
-Slowing the enemy so your teammate can get in some extra damage.

First off, tanks take a lot of damage before they go down, they can also tank towers if they need to. Zilean is not classified as a tank but building him as one works insanely well for your team. His ability to escape situations with his E is insanely helpful as it prevents feeding for you and your teammates.

Don't get me wrong, AP Zilean is very good, however when playing support it's very hard to gain AP quickly (compared to mid lane) since you have to build a support item, and you can't just steal your ADC's farm. So by the time you've built all that AP, The enemy has had enough time to build MR against you.