1. Don't play her if you aren't grouped with your ADC.
Playing Yuumi puts extra strain on the person you're attached to in order to work well. Randoms are less likely to consider your Q or R when they are positioning, being in a group allows you to more easily communicate to your ADC when to push up for your Q range. This is of course assuming you use voice chat...

2. Don't play her mid.
Her damage output is extremely weak compared to other mid laners. Unless your jungler is going to camp your lane, you're going to have a bad time.

3. Don't be alone
Similar to why you shouldn't play her mid, you should always be close to your team. She works best when she hops back and forth between allies to get better positioning and transfer bonus power. Detaching to give yourself a better position in a fight for your ult is fine, but don't play solo for too long, you are the squishiest champion in the game.

Yuumi only needs flash if she plays by herself. Her W is way better than flash since it has a shorter CD, longer range, AND makes her invincible. You shouldn't be far enough away from your team to ever need flash anyways, and if you're the only one alive after a failed teamfight, you don't have enough damage to do anything by yourself anyways. The benefits of taking a second supporting summoner outweigh the loss of flash's utility.

The ideal order for spells is R E W Q.

R Yuumi's ult is a powerful teamfight tool and can easily lock down the whole enemy team if you use it well. If you take rylais or twin shadows, their slow makes it extremely difficult to dodge the root without blowing an escape CD. Just be sure not to die while ulting....

E This heal is pretty sizable, especially when you store both charges to use at once. Using it to proc the Ardent Censer attack speed can easily allow your partner ADC to deal a bunch of extra damage in a fight. The speed boost can also allow you and your partner to escape death, or secure a kill.

W The donated power is very nice at later levels, especially if you have a ton of AP. Just don't jump off your partner by accident.

Q Yuumi's scaling on this spell is terrible. Even at high ranks with a bunch of AP, it only really hurts when it reaches its max time while attached to your partner. As such, this ability shouldn't be really used specifically for the damage, but more for the slow and spellthief's proc.

Since you should be spending most of your time attached to your ADC, you won't be running very often, so I don't suggest grabbing boots. Boots don't affect your partner's speed, or your W speed/cooldown outside of Lucidity. If you really want boots, lucidity boots are by far your best option. Tabi or Merc Treads only help a small build with your squishiness, but they don't come close to making you tough.

One benefit of being untouchable on your ADC is not dying. What item does more for an AP champ that doesn't die than mejai's? Turn that massive AP book into some decent AD for your ADC, or another large boost of AP to your Veigar. As long as your partner doesn't make dumb plays, you shouldn't die too often. (Of course, this is completely thrown out the window by CC-heavy bot lanes)

During the early laning phase, if your health is high, take advantage of your passive and spellthief's by auto attacking the enemy laners. You won't deal much damage at all, but the bonus gold can add up after a while. Hop off your ADC when you have 2 or 3 stacks and your Q is on CD, then hop back on when you're done. It also isn't a bad idea to hop off just to refresh your passive's shield.

ALWAYS hop off while taking towers. Your partner gets no indicator if you are targeted by a tower, and many won't notice until at least the first shot lands on you (assuming they even do notice). Plus, you can basic attack the turret to proc your spellthief's. 99% of the time, you'll deal more damage by basic attacking it yourself than you would provide your partner.

Attach when you get shot at. One or two small damaging spells are fine to take, but be VERY cautious. What seems like a small amount of damage can quickly rack up when you're as squishy as Yuumi. You especially need to hit W the instant you see a hook or other CC attack being lined up.

Jump on to ulting allies. If you see your karthus channeling his ult, or you happen to be near your Nocturne when he summons the shadows, you should get on them asap. You will provide a fair amount of bonus damage with your mere existence.

Outside of these situations, its hard to explain when you should jump on and off allies, its really down to experience.

As Yuumi, you become more powerful the more teleports are on your team. As discussed above, you should already be in a group with your ADC. When you see them recalling, feel free to join them if they have TP up. Popping on last second allows you to both buy items, and if they have teleport, you can get right back into the action without missing much, if any, CS. It's a powerful tool if the enemy bot lane doesn't have it at all, because you can turn any amount of gold into item advantage, and if they want to buy items to remove that advantage, they have to walk all the way back.

How about teleport on Yuumi? If you have multiple teleports on your team, not only can this be used to bluff your enemies into thinking a more powerful ally is coming in, but you can also use it to help out other lanes easily. Is your Jax top struggling against Gnar? Teleport on up there and give him some support. Once the deed is done, recall and run on back to bot. It's also useful if you somehow lose your ADC. Maybe they left shop before you were ready, or you missed your hop when they recalled/teleported? No worries, drop your own TP and you'll be back together in no time!

Got a mid or top laner running teleport as well? Great! Now if they end up needing help, just meet them in shop when you both recall and join them for the ride teleporting back to lane! Bonus points if your jungler ganks the same time you help out.

Hyper-aggressive enemies. Due to the fact that Yuumi is, indeed, not a body and can't take a hit at all, you won't have fun against aggressive bot lanes. You can't take hits for your ADC, or risk dying yourself, and you don't really have much CC to protect yourself or your ADC until later on. Your best bet is to sit back and play the waiting game. Working closely with your jungler to get frequent ganks is always a plus, but don't rely on them too much, they do have two other lanes to watch.

The best way to minimize this weakness is to attack minions as rarely as possible, allowing your enemies to push up a bit so their jungler won't dare gank you, and so you can easily fall back under your turret if you get attacked.