Bring Exhaust + Ignite. Double support summs gives you twice as many opportunities to secure a kill. You don't need Flash. You're more than mobile enough with teammates around, and without them around, Flash won't save you. You don't have any clutch Flash plays you can make, and you'll be spending a good portion of the game unable to Flash anyway.
Max Q -> W -> E
Rush
, then build
. Don't build support items, your usefulness is not your heal, it's the poke/slow, speed, and adaptive force, which all scales with AP, not heal/shield power.
Unsealed Spellbook is a great keystone on her. You can use your summoners much more liberally, since you can just overlap the ones on cooldown with a different summoner. Teleport to support your top or mid, Smite an objective, add another Heal for extra sustain to your carry, even Clarity can be incredibly useful since you blow through mana so fast. Summon Aery is still a great option, but Unsealed Spellbook has so much more impact. Free Boots, Future's Market/Biscuits, and Cosmic Insight are good choices.
Do not use your E to heal an ally unless they're below 50% HP. If they're above 50%, you should be trying to speed them into a trade or out of a fight. The heal is secondary, and won't do much until you finish Athene's. Focus on poking with your Q and setting up trades by manipulating the speed of people around you, not just healing/shielding.
Yuumi's job is to create numerous small advantages simultaneously that add up to big swings of power. Individually, the pieces of her kit are rather weak, but when you give a carry a boost of speed, extra AD/AP, a shield, and a heal all at the same time, while their target gets slowed and softened up, she can set up some major plays.