I just need you guys' opinions on item order. Which of the item order is best?

I've been going Rageblade then Trinity for survivability and damage. But I've seen some with Nashor first then Rageblade.

Could you guys give me 3 core items other than boots in order that is best for you?

Some items that I'm considering: Nashors > Aegis > Rageblade

I like to go Meki's Pendant -> chalice then rageblade, the mana regen is really necessary and you can easily get the chalice by level 5 if you need it. Allows much more spamming of everything to keep your lane partner alive and allows for activating RF to push creeps back/harass. Then Rageblade, its the bread and butter. I have been going for Hextech Gunblade after that, as either component is good for you, depending on what you like, and it allows you to do some very nasty activate RF > Snare > kill someone/weaken them badly for teamfights, basically it does a severe drop in their HP and teammates tend to jump on them as a focus.

Then again, I'm level 18 and in normal games, so people tend to need some... help in focusing haha.

Well your on the right track. Nashors and rageblade are pretty much standard gear, but its debatable (and dare i say situational) which comes first. Its hard to say on the third item without knowing what direction you wanna go. If you are going for a more beefy dps build, items with AS are really good as they synergize well with her E skill. Madreds bloodrazor and Last whisper could both have huge potential vs the right enemies. If you want an aura item, i would strongly consider strarks over aegis, as its more benifical late game unless someone on your team already has it.

A Kayle that swings twice a sec and that splashes both physical and magic damage can be hard to ignore. Don't forget the target of your Q also takes additional bonus damage.

On a different note, I have had some luck stacking aura items in the past as well. Soul shroud, aegis, and starks by 10-15min with your invulnerable makes for lots of very powerful teammates mid game, but it nerfs your dps output big time. I find with her new passive, other items work much better than they ever did before.

As for boots, it all depends on the team. Dont underestimate how effective a pair of BG can be if you get an early advantage. With enough AS stacked early, Kayle can easily push a turret by lvl 6 using E and her ult if a tower is ungaurded with just BG and AS marks. Once its down you have more breathing room to support other lanes and set up ganks, all which can lead to a huge snowball effect. Most of the time though, i get BS so i have the speed to jump in and heal when things go bad, and still have the speed to get away so i dont die myself. NT and MT are also great choices if you find yourself struggling vs a stacked melee dps team or a team/lane with lots of stuns.

I'm sure ill get hate for the BG comment, but dont knock it till you have tried it...it works, and before you say "why not just get stinger", its because stinger = no boots by that point. The early speed level 2, paired with your W is what helps you get around and escape the inevitable gank attempt on you as few have boots past level 1 by then. Otherwise you have no boots as well and cant run away. I don't endorse BG often, but i have used them and seen it lead to an early advantage with Kayle if the situation is right... ie they have an ungaurded tower with no teleport abilities.

Try and read some of the very well written guides on this forum for kayle. They usually have good advice on item builds.

Edit: I totally forgot about chalice, Chalice is golden honey on Kayle, and i highly endorse its use if you like to spam your healz all over the place and would rather not take clarity. Taking clarity as a summoner spell works almost as well though, and frees up your first item for something more juicy.

The core build for Kayle end game is,

Ninja Tabi or Mercury Treads
Guinsoo's Rageblade
Nashors Tooth
Banshees Veil, OR Zhonya's Ring

Summoner Spells: I use Ignite because its the new "OP" and Clairvoyance. Although, you can choose someting else.

Never get any other boots except for Merc Treads or Ninja Tabi. Kayle has enough speed with her heal, and she is more defensive.

Runes:
Red: Attack Speed, or Magic Pen
Yellow: Mana Regen per 5 levels
Blue: Flat Cool down Reduction
Quints: Flat HP

Now Zhonya's or Banshees Veil?

Banshee's Pros and Cons

+Added Survivabilty
+Bubble makes you less targetable
+You get a Catalyst early game, which is a VERY good item
+More Tanky

-A LOT Less Damage

Zhonya's

+More Damage
+Active Abilty gives more survivability if you use it right

-Less "true" survivabilty
-Only Active can shield a spell. Banshee's is on a cool down of 30 sec. Zhonya's is 1 minute