Hello, I'm Eohdorr, I main Orianna, and this is what I do with my Orianna. This may not be the best Orianna build, and it might not work for everyone, but this is my reasoning for my items, so I hope you can understand where i'm coming from. :)

Normal Start: This is a standard start I use. Orianna becomes quite mana hungry in the early parts of the game, and this helps you with it. This is also what most mid laners take, and I use it in 90% of my match-ups.

AD is scary Start: I personally HATE this start, but it helps you get your Zhonyas faster, and help you survive against one of Orianna's hardest matchups: Zed. This is literally the only time I use this start. Be VERY cautious with your mana, use it for your E and Q to get those outta reach minions, play this lane smart.

First Back (A): Orianna needs Mana Regen. Build for an Athenes. This is for those mid match-ups that hurt. Such as Syndra, Leblanc, etc. People that hurt early, or have nice burst potential. However, if you feel you don't need to worry about their burst, or you feel you need more power instead of that MR, then use First Back (B)

First Back (B): Orianna needs Mana Regen. Build for a Morrellonomicon. This is for those match-ups that you don't fear the enemy. Infact, you want more power to dominate harder. Or maybe they're a Lifestealing/spellvamping lane, and you want to hinder that. Or maybe its the composition you're worried about. maybe their team has a lot of heals. I take this more than not. It gives more damage than Athenes, gives slightly less mana regen, and gives same ammount of CDR. If you don't know if you want to go with first back A or B, and feel your okay for a little while without mana regen, just take Codex and wait to see how the lane plays out.

Core Items: MUST HAVE: No reason NOT to get these items as orianna. Almost no reason at all. Rabadons for AP, Void staff for Magic Pen, and sorc boots because Ninja tabi doesn't help you much. Merc Treads maybe, but there are plenty of MR options. If you're looking for tenacity, then why are you playing a safe mid like Orianna? :P
But again, I won't stop you from building it, your choice.

Armor/Anti-AD Items: Let me explain each item. Don't build ALL of them, decide which ones take priority.

Zhonyas: 50 armor, and a stasis that'll save you more than not.

Morello: This is odd to take, but it's a CDR item and it helps against Lifestealing ADs. Athenes wont save you from physical damage, no mana regen item will aside from Seraphs, which i personally hate on orianna. But if you want to build it, I won't stop you.

Rylai's: This gives you a lot of health to survive, and the slows to help you escape. Not as good as zhonyas, so it definatly shouldn't be a replacement for it, but if you have zhonyas already and you're still dying, well maybe this item can help. Can also replace GA while its on CD. (Lategame only)

Magic Resistance/Anti-AP Items: Again, I'll explain each item's reasonings. Again, decide your priorities.

Athenes: Gives MR, gives mana regen (much needed) and gives AP along with CDR. How does this item NOT cross your mind? :P

Abyssal Scepter: gives you plenty of MR along with aggressive AP, fits orianna perfectly because of her teamfight capabilities. Not something you want early, however. If you want something early, consider Athenes and Banshee's.

Banshee's: Gets rid of a spell for a bit and gives hp and MR. Really solid item. Not something you want early game exactly, but if you're losing hard, go ahead. Not a bad item either lategame.

GA: Solid MR item and revives you once. Replace it with Abyssal/Banshee's if its on CD late game.

No Fear build: Full AP build, No fears, just going standard AP.
Rylias and Liandries give some HP, but still offer massive damage, and its not like that HP hurts you. Besides, your an AOE teamfight champion, that liandries and rylai's speaks for itself during teamfights.

Can't explain much here, but i'll try. Your Q is your central source of damage output, and all your other abilities revolve around it, so its essential to max it first. You can level your E as your first ability if its a touch lane for you. (Such as Leblanc or Zed) If you do so, then do: E - Q - W - E - Q - R. Then follow the rest of the skill order. This essentially replaces the 2nd point into your W you get before your ult. Makes it slightly easier to survive lane. I wouldn't put too much more into it than that, however.

Your combo should be QWE. Fire your Q behind them, then W quickly and E so the returning damage from your E hits them. Try to get in one auto attack while your E returns to you. This'll help you defend against minion attacks and reduce the damage you take, while dealing the most effective amount of damage to the enemies.

A lot of people will tell you to ball them during teamfights to get your ult on someone. This is effective in only 2 cases:
1: you want to burst the hell out of that 1 person before he can do literally ANYTHING. Make sure they have a strong lockdown (Malphite, Vi, etc.)
2: if they have a strong aoe teamfight ult, and they're in tight spaces such as the jungle, so its 100% reliant on their positioning for your ult to do good. You'll hit 2 or 3 targets.

Me personally, I ignore them. A lot of people want that ball on them because they see wombo combos and they want to do it too, or they think its the best way to use your ult. I use my Q instead to position it. If you hit 4 people at least, that ult will around 75% better on average. I've won and lost teamfights due to this way of thinking, but I feel I've won more teamfights than lost due to this, because the person who wants me to ball them usually only hits 1 person. Do this only if you feel that 1 person NEEDS to be eliminated. such as a REALLY fed ADC/mid.

Quints: Flat AP or Magic Pen
Both of these are good choices, I like flat AP because it works better for you in the early game. Late game the Magic Pen will be better, but you decide which works best for you.

Reds: Magic Pen or Hybrid Pen
In my experiments, the Hybrid is not common, but it deals more damage. Especially with Orianna's passive. It's a bit of unexpected damage. Go with either or, Again, YOU experiment. YOU find out which works best for your style of game play.

Yellows: Health
Better than armor, you want something to defend against minion dmg. No other option here imho.

Blues: MR or flat AP
Magic resistance is majorly important, but as orianna you get MR from your E. So if you want to be more aggressive with those blues, go for it. I personally like MR.

I can add more if anyone asks me for my thoughts on others. REMEMBER, THIS IS MY VIEW ON THESE MATCH-UPS. ANYONE CAN BEAT ANYONE, IF THE SKILL GAP IS HIGH ENOUGH. Play to your champion's strengths and their weaknesses. Ult wisely.

Zed: Personally this seems like the most deadliest match-up for Orianna, even worse than Syndra. Orianna's poke can't compare to Zed's, and no mobility to evade the ult.

Syndra: She outranges you, she has slightly more damage than you, and have a stun that can immobilize you easily. Tough anyway you look at it.

Ahri: Many tell me Ahri beats Orianna. I personally beat Orianna as Ahri, and beat Ahri as Orianna. Not every time, but I find it a skill matchup. Orianna can out poke Ahri, But Ahri can easily evade your ult. Play this smart, and ult when she doesn't expect it. Keep in mind she can ult dash and charm you, so save your ult until she is either on you, or her ult is run out. If she is on you, E yourself and ult carefully.

Annie: One of Orianna's easiest match-ups. You outrange Annie in every way, and you out damage her. Just don't let her stun you, and you'll win.

Yasuo: Yasuo is annoying, and beats a lot of mids, but oddly Orianna doesn't lose like the others. Yasuo's windwall can be useless if orianna's ball is in the right spot. Play this wisely, you have no escapes.