Luna is a carry hero with a strong presence at all stages of the game, initially thanks to her magic damage abilities but, later on, as she acquires items, through AoE physical damage output.
Skills
Skill Lucent Beam at level 2, but make sure you max it by level 7 as it determines the power of your ultimate.
Moon Glaives is skilled and maxed last.
Lunar Blessing is skilled first to help with last hitting and maxed second.
Eclipse is always skilled (levels 6, 11 and 16). Use to get pick-offs and in teamfights to deal a lot of magic damage.
Items
Starting Items: A set of Tangos, a Healing Salve, a Circlet, 3x Iron Branches.
Early Game Situational: Hand of Midas (greedy, having an amazing early game), Helm of the Dominator (slightly less greedy, having a good early game), Drums of Endurance (poor or early teamfighting), Ring of Aquila (cost efficient stats).
Early Game Core: Power Treads, Magic Wand, Wraith Band.
Mid Game Core: Black King Bar, Manta Style.
Luxury: Satanic, Butterfly, Monkey King Bar, Daedalus, Heart of Tarrasque, Divine Rapier.