Use Moonfall and Lunar Rush to stay on targets and activate for extra damage.
The time it takes for Crescent Strike to reach your target at max range is the same time Lunar Rush takes to reach so if you cast Crescent Strike into Lunar Rush you will get the cooldown refresh immediately.
Diana is extremely agile in the jungle due to the synergy of Crescent Strike and Lunar Rush. Use them in conjunction to traverse the map quickly or to escape enemies.
Lich Bane works very well on Diana due to her low cooldown on Crescent Strike and her ability to quickly close the distance with Lunar Rush, and will give a major boost to your burst damage provided you have built enough AP items. The extra movement speed will also allow you to stick to your target better.
Even when building for survivability, make sure to buy items which also offer [AP] to maximize damage output. Abyssal Scepter will give a good boost to your magic resistance, whilst Zhonya's Hourglass should be built when the enemy teams deals a lot of physical damage.
Diana can become deceptively durable with a Will of the Ancients due to the low cooldown of her damaging abilities.
Dodge Crescent Strike or move to safety if you are affected by Moonlight.
Pale Cascade orbs only last a few seconds. Avoid Diana and engage her after the shield dissipates.
Diana can play very aggressively if she uses Lunar Rush without Moonlight, but you can punish her by slowing or stunning her when she has no way to get back into position.
Diana is able to kill squishy characters easily, even under a turret due to her shield. Spending on an early Hexdrinker is a good option for all AD carries to build which can later be changed in a Maw of Malmortius.