Ignite deals some nice damage. If you determined to kill, this would be best choice.

If you want to assist your teammate or pick a free kill, bring Teleport.

If your opponent has deadly killing ability in early game, Exhaust will be your savior.

Since Diana has nice sustainability with her Pale Cascade, I pick up Greater Glyph of Ability Power to increase her damage. However, you still needs Greater Seal of Armor because harassment from basic attack will be very often in early game.

You need Crescent Strike to reset the CD of Lunar Rush. You need Lunar Rush to make your Moonsilver Blade and Moonfall work well. Thus your core skill is absolutely Crescent Strike. Spend points on it to reduce its CD and things will be smooth all the way.

Upgrading Pale Cascade can increase the sustainability from it, which is better than a higher Moonfall.

STARTING ITEMS
Diana is not a bogey before Lunar Rush. Champions with strong poking ability can crush you in early game. You need to survive.

Lich Bane
Your Lunar Rush will make you start basic attack on your target right after dashing. Moreover, the damage from its passive is more than that Abyssal Mask can provide in most situations. You can take a calculator to verify that. The advantage of Abyssal Mask over Lich Bane is MR only, so I buy a Negatron Cloak to make it up. If you are facing an AD champion, you should pass it and rush Seeker's Armguard before Lich Bane.

BOOTS
Diana needs her Crescent Strike to launch an assault, so the Ionian Boots of Lucidity fits her well. If you feel you will be focused in fight, you should take Mercury's Treads instead.

Zhonya's Hourglass
This provides plenty AP and also sustainability for 1.5 second. The armor works good with your Pale Cascade, too. Really a great item which strengthens your ability both offensively and defensively.

Abyssal Mask
This is upgraded from Negatron Cloak. If you didn't buy it according to your enemies, you can take Liandry's Anguish or Void Staff.

Nashor's Tooth
This fits your Moonsilver Blade very well and can give you strong persistent damage-dealing ability and potential to BD. You can consider it when your enemies start stacking HP.