CORE ITEMS

Your first item will 99% of the time be Athenes. This synergizes very well with Yuumi's Q for dmg and her E for healing. This is the power spike for Yuumi

After getting Athenes, you need to check enemy team comp.
Do they have a lot of CC?
If yes, then buy Crucible ASAP. UNLESS it is a suppression (WW or Malz ult), Airborne (Yasuo, Cho, etc), Nearsighted (Noc, Quinn, Graves), Blind (Teemo, Quinn), Disarm (Quinn) , Ground (Cass, Poppy, etc).
If no, then get ardent.

If it doesn't seem like you will die or you are ahead, upgrade Dark seal to Soulstealer.

Optional DMG Item build after:

Do you need damage and slow? Get GLP.
Do you need burst damage? Get Ludens.
Do they have lots of healing? Get Morellos.

Optional Support Item Builds:

Get Redemption if your team has a lot of bruisers or tanks that can last long enough in fights to make use of Redemption.

Get Shurelyas for assassins or more mobility for your teammates. (need to dodge lots of spells, run away faster, chase, etc).

LANING PHASE

In laning phase, you want to mainly:
1. Heal ADC with E when they are below 70-80% HP (Or less depending on your mana)
2. Get out and pop passive for Shield and Mana
3. Poke with Q if you have spare mana or don't need to heal as much.
In that priority order.

MID/LATE GAME

You want to stay with your ADC mid to late game and keep them alive while they deal a lot of damage. If your ADC is really behind (you had a really bad laning phase) or you can't save him from dying, then you can roam with jg or with the strongest person on the team right now. But be careful not to stay with them for too much because you can end up leeching exp from the laners and putting them behind. Only stay with them until your ADC farms up enough and become strong, then rejoin your adc.

TEAMFIGHT

In teamfighting, Yuumi should stay with the frontline and ult as many people as possible. After that (or during), use W to go between people to get as many people in the ult as possible and to heal the low ones. DON'T forget your ADC because they are really squishy and as soon as you leave their side they can easily die. Mostly stay with your adc after your ult run out unless someone is low then heal them and go back to adc.

Yuumi's PASSIVE

When facing against a high CC champ, make sure that they have USED their CC ability, or bait it out, before walking up to proc passive.
If they don't use their CC ability, sometimes it is better to just stay inside your adc and not get out until the middle of a fight than to risk dying. Because Yuumi is super squishy, 1 cc can mean her death, especially if the enemy bot lane is able to chain cc.

In a fight, always go out to proc passive so that you can shield your adc as her passive is quite a good shield. However, always keep in mind the CD of your W when you pop out, because sometimes when the passive CD is down and you just W to your ally to heal them, you W out for passive and can't get back in, leading to both of you dying.

In teamfights, remember to keep using your passive. Make sure that the enemy has used up AOE CC first. In some enemy team comps where they have a lot of CC, it is better to be safe by not going out at all and just staying in your allies and healing them. That is more useful than just dying.

Yuumi's Q

You want to use Q in lane mainly to poke. Try to target the side first instead of straight forward so that it can proc the 1 second extra damage.
It is easy for the enemy to shield or dodge Yuumi's q by going inside minions so keep that in mind. Try to use Q when there is no minion wave (usually you want to use it when there is only ~3 minion left in lane so that it is hard to dodge).
Also, it deals more damage when the enemy has high health so when you get back in lane and everyone has full health try to use it more.

When chasing, you can also use Q to slow the enemy since it has a 20% slow in its empowered form.

In teamfights, try to twirl your q around so that you hit people with empowered q, not just normal q.

Yuumi's W

In laning phase, this can be used to proc Aery, giving your Adc a small shield. This is especially useful if they have Comet and you can use Aery to shield your Adc from the dmg of their comet.
In lane, you will also use W mainly to proc your passive.
NOTE: It is important that when you are with an ADC with an escape (Ezreal, Trist, etc) you don't use W to proc a shield in the middle of the fight until AFTER they use the escape ability. Otherwise, you will just be stuck in the middle of the lane with a W CD while they escape and then you will either die or your adc will come back to save you and you both die.

Use Yuumi's W mainly to redirect the healing to an ally who is low, or to save yourself when your ally is going to die. Yuumi's W will go on CD if you just attached to your ally and then they die, leaving you vulnerable.
A good Yuumi should know when it is possible to save an ally and when they cannot be saved, so you should let them die and save yourself. However, if your ally has an escape ability, usually your 1 heal is enough to give them enough health to escape, so take risks.
You should also take note of the CD on your W everything you detach especially for passive. This is very important because your W basically decides how vulnerable you are so timing is important.

Yuumi's E

Yuumi's E is her main kit as she can literally be just a healbot. In laning phase you want to use your E not only defensively to heal your Adc but also offensively like chasing the enemy under tower for 1 autoattack or small trades because Yuumi's E also gives attack speed and movement speed. It can also be used to dodge abilities. Yuumi's E also procs aery so you can get a heal+shield from it. Sometimes that shield is enough to win trades when using E offensively.
Try to use Q first before you E when you have Athenes because that will make your heal a lot stronger, around 50-80% more heal.

In teamfights, you want to use W and E together to heal ALL of your allies, but you must prioritize your ADC. Yuumi is good in that she can heal everyone and not just one person. Her E cooldown is also pretty low late game.

Yuumi's R

In laning phase, sometimes when your ADC is CCed you want to pop out of your adc to get a better angle in your ult. Try to hit both enemy botlaners with your ult as it deals some good damage too.
If Yuumi gets CCed, it will STOP her ult. So most of the time unless they have CC not on CD you don't want to get out of your ADC yet.
When Yuumi is ulting, she cannot proc her passive, so keep that in mind.

In teamfights, use W to root as many people as you can with your ult.

ABILITY CHOICES

Generally, we want to max R->E->Q->W
I like to alternate my Q and W in the middle of the game because it seems more effective. Mostly Q will be used to deal damage for Athene's extra healing.
E must be maxed first because Yuumi can heal a lot of health, it's pretty cancer.

LEVEL 3

Taking Q if:
1. My ADC is NOT low.
2. WE are poking them and they are lower than us. It doesn't look like a fight is going to start soon, and even if it is, we will probably win.
Taking E if:
1. My ADC IS low
2. We are being poked into tower or unable to trade back because I have to constantly heal my adc and it's taking up all my mana.

Late Game Q and W choices

I like to alternate between Q and W late game after I have maxed my E out. However, if you think that your Q isn't doing enough damage because of that, then you should max Q first before taking any points in W. It is situational and depends on the enemy team comp.
If you are going offensive build it is sometimes also better to max Q first.