(1) I've updated the build to be a bit better. I over estimated the helpfulness of the Archangel's Staff. I've altered the items accordingly.

(2) Due to overzealous cooldowning, I've changed up the ending items to add more AP and Magic Penetration.

Orianna can be devastating and not to be taken lightly. This is the build i like to use as it allows for rapid mana regen early in the game while maintaining cooldown reduction to ensure rapid combos.

Greater Marks of Insight for that Magic Penetration.
Greater Seals of Warding to buff early ability hits.
Greater Glyphs of Potency for the AP.
Greater Quintessence of Insight to really drive home that Magic Penetration.

I went 9/0/21 with these. Enough to get the magic penetration and the scaling AP.
I put emphasis on the mana regen as well as hitting Spacial Accuracy and Insight to compliment the 2 summoner spells.

This builds rather fast building and don't be afraid to go back to base often (when the opportunity arises) early on to get the next item.
By choosing the Meki Pendant, I open up the chance to grab potions early on. I grab a health and a mana to make sure I can stay in the lane as long as possible. Even with Clarity, I find myself draining mana fast after I get my second and third abilities. You many need to keep up with the potions (I go 2 of each) on the second or even third run to base before you get the next item.
Philosopher's Stone give both mana and health regen as well as that extra gold. this is a really good early item as it helps keep you in the lane.
Ionian Boots of Clarity are next. It seems like a late entry to get boots, but with Teleport, you don't have too worry too much about the walk. Also, with Dissonance you can get back fairly quickly on your own.
Moonflair Spellblade gives you a bit of AP and that tenacity to help get you out of fights and ganks.
Next, cash in that Philosopher's Stone for Shurelya's Reverie. It'll get you a bit more health and more regen as well as a bit more cooldown.
Grab a Will of the Ancients next to help your teammates out followed by a Rabadon's Deathcap to get your effectiveness up. Add a Void Staff to tear through enemy champs.

If I'm having trouble with dying because I've over extended myself, I'll hit up a Frozen Heart for the armor, mana pool, and slowing effects. If you have a particularly hard hitting Caitlyn or what have you, fitting in a Thornmail might be a good bet too.

Start with Command: Attack for some early harassment, but really focus on Command: Dissonance. I usually grab Command: Protect on level 3 to return the ball quickly if I need to make a run for it. Once Dissonance is maxed, focus on Attack unless you are getting a lot of attention and need that extra shield to help you out.

I stick with Teleport and Clarity with Orianna. Teleport will get you to the lane fast to clear creeps or act as support. Orianna is great as support but she can handle solo mid just as well.

Clarity will help keep your mana going and stay in lane even in late game. she uses alot of mana but mid to late game she doesn't rely on it so much.

Orianna can do great anywhere, but she really shines when teaming up. I like to keep her with a melee champ so she can hang back and harass while acting as support as well with her Protect and Dissonance. Her ball acts as another laning partner and you can confuse your opponents by flinging that ball from an unpredicted direction. They'll see me go into brush behind my ally and find themselves getting hit from behind by the ball, usually with a slow following.

If you're in mid, use the tower to your advantage, especially when you get your ultimate.

Don't worry about going head on into battle for kills. While she can take on a squishy opponent, I usually find myself getting the kill as I'm getting chased, see it's only the one, and turn around with a full barrage of my abilities. taking them down to very little health very quickly. It tends to be more of a gamble, but I can usually still get away if I screw it up, or take them out with Clockwork and a well placed Command: Attack if they think they can take me down first.

Tend to focus on the creeps when you're not supporting. You'll be too squishy to really go 1 on 1 with most champs. This will also help your team gain tower territory. She can be good at back-dooring but always be watchful of the gank. If you manage to get a tower down fall back as you'll usually have 2 or 3 coming at you. Orianna is great at fleeing from ganks.

She's not to great jungling for obvious reasons, but mid to late game, if you can manage to get away, grabbing a buff or 2 shouldn't be too hard to do.

Orianna can be a great asset to any team if you know how to play her. Don't be greedy and don't expect a lot of kills, but do expect assists. She can save a teammate from death with a protect and dissonance combo. Assume the role of support when you're able and blast those creeps when you can't.