Startup Dota Items For Luna
If nobody buys a chicken, get one. It will serve you well because you should always try to stay within grasp of gold and exp, and you are always one step ahead of the enemy who doesn't have one. Grab 2 branches, and spend the remaining gold on tangoes + clarities. Stats and health regen are basic needs while clarities will provide the necessary mana regen for you to use Lucent Beams. Share the chicken, and ask for a contribution to upgrade the chicken.
Why HotD?
For 1850 gold, Helm of the Dominator gives Luna a usable orb, which is lifesteal. Lifesteal helps to boost her field staying power so she doesn't need to go back to fountain. Going back to fountain means loss of precious time to farm for exp and gold. HotD also has the active skill Dominate which lets you to dominate a creep under your control. This can be used to Dominate a neutral creep to tank for you and it can also be used to assist you in creep stacking. In the buildup, you should always buy Helm of Iron Will before Mask of Death, because helm gives you +3hp regen and +5 armor, which is always better than 10% lifesteal where you don't attack often, as you mostly aim for last hits. As a bonus, HotD also provides 20 damage, which is always a good thing.
Why Power Treads?
Boots of Travel is good if you get easy games and lots of farm, otherwise just stick to Scroll for teleports or changing farming lanes because 2200 gold is not a small amount.
As of 6.63, Phase Boots now gives +30 IAS as well, but losing the armor bonus. Seriously speaking, Power Treads is still better. Why? This is because Luna doesn't need the active ability Phase. She has one of the highest base ms of 320, and with the new Treads it's already 390. It's also 500 cheaper now, so you can get the stat bonuses earlier, either for survival, for Lucent spam or just for the +13 AGI. +13 AGI already gives +13 attack and +13 IAS for Luna. Phase Boots is not worth it for the +17 IAS more compared to the flexibility of Power Treads.Toggling from Str/Agi/Int will work for Luna in terms of surviving/attacking/casting.
Why BKB?
Killing Luna in the shortest time possible is the only way to get rid of Eclipse, and the only way to do this is by using nukes. BKB's precious few seconds of magic immunity will let Luna unleash a total Eclipse without interference. You can already enjoy the HP boost as early as when you get its buildup part, Ogre Axe. BKB also further boosts her damage by 24. In lower level games where there are less co-ordinated fights, you can actually skip BKB and go straight for a Manta Style. The farming ability that Manta Style gives will skyrocket your gold compared to your enemies.
Scroll of Town Portal
Luna needs it, so remember to carry one at all times, it's cheap, it's worth it, and you'll never know when you'll need it.
Bracers
Bracers provide cheap Str and some stats. Get them if you have trouble surviving, or Wraith Bands for some stats and early dps. If you can rush a Belt of Giant Strength and Ogre Axe though, you can skip them as you wish.
Yasha
You get extensions if the game doesn't seem to be ending even after you've gotten your core items. If the game is difficult and farming is tough, you may consider getting Yasha. It boosts Luna's dps and gives a 10% increase in movespeed, which makes her have 418 ms, when equipped with Power Treads together. Speed helps to get her in position for a good Eclipse, and contributes in both chasing and escaping.
Manta Style
Manta Style is a cheap and good choice if you managed to get some kills and decent farm. It's a powerful item against single targeted spells since it gives you images to confuse enemies. The stats it gives are all very useful for Luna, boosts her dps and survivability at the same time. Its active skill Mirror Image further boosts her dps with illusions, which are excellent in pushes and combats. It can also be used to dispel some negative buffs off you. The best thing however, is that Moon Glaive works with illusions, which means more bouncing glaives, more dps to multiple enemies! Just remember that casting Eclipse/BKB with your illusions around will give away your identity, although in most cases enemies won't have enough time to retaliate against your combo.
Examples of total DPS boosting (rounded numbers)
Max Moon Glaive only
Base damage + stats = 100 damage
Total including Moon Glaive = 252 damage
Max Lunar Blessing only
Base damage + stats = 100 damage
Total including Lunar Blessing = 127 damage
Max Moon Glaive + Max Lunar Blessing only
Base damage + stats = 100 damage
Total including Moon Glaive = 252 damage
Total including Lunar Blessing = 320 damage
Max Moon Glaive + Max Lunar Blessing + Manta Style's Images
Base damage + stats = 100 damage
Total including Moon Glaive = 252 damage
Total including Lunar Blessing = 320 damage
Total including Manta Style = 531 damage
The Butterfly
With the farming ability that Manta Style is able to give you, you should have no problem getting The Butterfly if the game drags on. The Butterfly increases your dps tremendously as well as boosting your effective HP. Since it boosts your AGI, it reaps more bonus from what Lunar Blessing and Moon Glaive can give. It further strengthens your role as a carry by being a multi-unit dps monster in a team fight. In an impending team fight, you should always time your entry. Luna doesn't have any escape mechanism so if you're going into the fray, make sure it counts and timed wisely, because once the enemies see you, they will focus fire on you to eliminate your dps threat. Try to anticipate combo spellcasters like ES or Enigma and let them exhaust their combos before going in, or else use BKB.
Satanic
Satanic is another extension that you may consider getting if you need more HP. Other than boosting your passive Lifesteal, its active skill that boosts your Lifesteal to 150% is no joke. If you're not disabled, it's practically a free heal for you as you're fighting. Your Manta images also benefit from the HP boosts since it's based off STR.