I'd say Zilean might be one of the most unique Champions in the game. Who else can actively level up their teammates? Time in a Bottle is a very useful passive ability, and being able to revive someone with Chronoshift can really turn the tide in a battle. He can be fairly tricky to master, though, since utilizing him relies a lot on landing Time Bomb, knowing when to boost a teammate, and who you should boost or revive, along with the fact he doesn't have any way to naturally shield or heal anyone before getting his Ultimate.
Early on you might get a lot of flame from your ADCs if they're not familiar with Zilean. They're expecting you to heal them, but Time Warp is useful for letting your teammate chase or escape, so they can't play as if they're with Yuumi or Soraka. Something people often got mad at me for when I first started playing Zilean was "kill stealing" whenever I got two kills. Since all of your damage comes from your time bombs and Ignite (if you choose it) you can't really help if you happen to kill enemies.
Crowd Control can be the biggest threat to Zilean, especially Stuns and Zoe's sleep ability. It can be worth to run Precision (specifically as secondary runes) for Legend: Tenacity against Champs with good stun abilities, like Leona and Veigar. Don't automatically take them only because you're against a support and/or adc with a stun ability if there are other Rune builds you prefer, some stunning abilities are easier to avoid and/or don't last as long as others, so champs with those might not be worth taking Tenacity for if you don't like it.

Zilean's Q Time Bomb can provide an Area of Effect stun if he lands a second one on the same target. This is the essential purpose of his W Rewind so when in range of an enemy champion, press Q-W-Q to stun them. Be mindful of trying to do this if you have just used your E Time Warp because Rewind can only be cast when one of your regular abilities are on cooldown. When you use Time Bomb beyond its range, Zilean will move to be in its range, so hitting Q-W-Q by habit will use Rewind for Time Warp and you won't be able to throw a second bomb.
Time Bomb prioritizes certain targets, so if it lands on a group of different things it could cling to, it will choose one. Number one priority is anyone who already has a bomb stuck to them since this activates the stun. While I may be taking a lazy way out, aside from knowing champs are a higher priority than minions, and enemy champs are essentially the second highest priority under "anything with a bomb on it", knowing the whole priority list probably won't be too helpful in general. Perhaps it can be helpful for advanced straps, but I'm trying to keep this more beginner friendly.
Time Warp can be used to speed up allies but it can also slow down enemies. There are times when using it on an enemy will be more useful than boosting an ally. For example, if a teammate is around the corner and you're in range to slow an enemy champ, do it then stun them with two Time Bombs so your teammate can come finish them off, plus you'll have used Rewind so you could slow the enemy again or speed that teammate if you need to. Another trick I like to do is when someone tries to take a shot at you or your adc by stepping into your turret's range, slow and/or stun them so they keep taking turret shots.
Zilean's ult Chronoshift can be very difficult to land. The general tip is to not wait until the last minute to cast it, because 5 seconds is a fairly generous window. Just so it's clear, your ult will place hourglasses around whoever it is cast on and if they die within 5 seconds they'll be revived. It can be hard to judge if someone will die within 5 seconds, sometimes you'll see someone lose a ton of health instantly, use it, then they suddenly won't take as much damage until your ult has ended. Sometimes the reverse happens where your teammate is fine but then all of their health is suddenly gone. While definitely don't wait for someone to be down to a tenth of their health, just use your own best judgement.
Something to keep in mind with Chronoshift is that the amount of health someone gets when they revive is boosted by 2x your ability power. With that in mind, the other Champions on your team may determine which items you want to buy. For example, you might want a lot of AP if you have a tank on your team, but this depends on whether or not most of their defense is in their HP, in which case you might want to save your ult for someone who can either tank more hits or can do a lot of damage. This is also why I think Kai'Sa can be a strong pair for Zilean because she's quite frail but can do a lot of damage, plus her Q and E should be ready by the time she revives, so she's likely to come back with nearly full HP and can unload on nearby enemies.
Time in a Bottle stores EXP and you can use it on a teammate once you have enough for them to level up. You can't just give them extra EXP whenever you want. I'd recommend waiting and letting it build up until you can get your adc either to level 6 or 3. If you and your ADC are having trouble early on, then you might want to get them to level 3 first because you're likely to level up with them at that point and you'll both have your 3 abilities ASAP. There's not much of a point leveling them up from 1 to 2 because by the time you can do that they'll be one minion away from leveling up. Time in a Bottle has a 2 minute cool down which is why it can be helpful not to use it whenever it's ready early in the game.

As of now, I don't think Zilean is as good of a choice as a support like he used to be. The new items have created a snowball focused meta right now, and that can be a bit difficult for Zilean. His kit is supportive, but his Q can cause problems. When the first to snowball, which usually means the first to get their first/mythic item, wins, such a big damaging AoE like Zilean's Q (especially when 2 are thrown) can easily result in stealing CS from your adc, which could then result in the enemy adc coming back with their mythic first, and it's difficult to get ahead of them or beat them in fights after that.
Mid lane might now be a more viable option for Zilean, or alternatively, he could make a fairly strong APC, where you don't have to be as mindful of your bombs and have more room to be aggressive.
I think Zilean's in a bit of a weird spot, or maybe he always has been and I'm only realizing it now. Only one of his abilities deal damage, and that's his main source of damage. That damaging ability is also how he stuns enemies (should be obvious I'm talking about his Q) and by landing 2. It's not quite enough to win 1v1s unless the fights are prolonged by quite a while, which makes him somewhat difficult in a solo lane, but it's also so much damage that it can be hard not to steal kills from your adc as support. However, Zilean is a champ that is just always useful to have on your team, especially if you're good with him. Leveling his allies and himself up faster is a useful tool only he has, and getting level 2 or 3 before your opponent can be a huge advantage. Obviously, his ult being a free Guardian Angel is so helpful, especially since it has a fairly low cool down.