For support Zilean, this is usually the build I aim for.
OR
. Boots are depending on what I need or if I might have more CDR elsewhere in the build. There are still other items you can use for whatever you need. Like if you want Liandry's for a bit more peel and health, Mikael's for the CC removal, etc.
As just about all supports, Try to get at least 800 before your first back so you can buy Sightstone. Anything else, put either into getting first stage of boots and pots or getting your Tear. Recommend getting Tear as soon as you have the gold for it after to help with mana. Use E only when you feel it can help secure a kill or to escape. I feel it's not worth the mana to use at times unless it's 2-3 times to leave base after backing to help stack Tear(only use W once to save mana when doing this AFTER the E wears off for efficiency).
Bombs are easy enough. They're just flat out better to me than old Zilean, and honestly, it makes him feel so much better despite his other changes. He's a much safer champ than the telegraphed "I'm moving close to bomb you!" which could get you in a bad spot. They can be used to zone out enemy laners really well. Try not to use double bombs too much unless it's to punish bad positioning or if enemy gets aggressive to save mana. One bomb is usually enough to deal easy poke or zone them away. Just try to get them to stick to champs rather than minions to avoid harming your ADC's CS.
When grouped up for team fights or shoving lane, it's really not that much different. Bombs away(hopefully for great stuns too) and try to hold onto your E to save a life or doom an enemy. Your R is more on what you feel is the best choice. Try to hold onto it until you KNOW it'll be useful and not just bring someone up to die again(unless that's what you aim for and helps secure a kill or two for your side as well).
I don't play Zilean all that often, but he is a champ I really like and is simple to understand. Just takes time and practice to learn how to use his skills effectively.