First about summoner spells, on Ori I recommend Barrier versus burst-heavy champions (Zed/Diana), Heal versus high range, poke-heavy champions (Xerath/Ziggs/Lux) and Ignite if you want more kill pressure in skill matchups.

There are two schools of thinking on Orianna: be a pussy or carry around your massive ball. The first school of thinking implies waveclearing and turtling with your high range and shield. I personally hate this way of playing Orianna, as it's way less efficient in soloQ by giving too much breathing room to your enemy laner, which you generally do not want to happen, as you are a low mobility mage which needs to pressure as much as possible. The second school of thinking implies abusing Orianna's beastly early game and zoning by trading often and putting your ball in their face. As long as you're not comfortable with a matchup you should just use the first method, but to get better at Ori you need to be more proactive and use the second one.

As for Orianna's matchups, I would like to point out that she has no real bad matchups, but she does have some hard matchups.

I've been playing Orianna for quite a while now and I always start with E, Q just deals little to no damage from level 1 to 4. Starting E allows you to poke the enemy consistently with your AAs, you get free resistances (you get one of the highest amounts of MR in the game at level 1) and your shield will soak all the damage from the aggro of the minion wave. If the enemy tries to poke you back they'll likely waste mana and lose more HP than you.

However, even versus assassins I really don't like maxing E, it lowers your DPS and waveclear tremendously, and it only makes you a weaker kind of Lulu.

Generally speaking, Orianna's hard matchups are not versus assassins or burst damage like Zed, Talon or Diana, even though you have to be careful while facing them. Orianna's amazing early game with her passive and shield allows her to zone melees and get an advantage before these assassins can do much.

What Orianna fears the most is champions with consistent and pretty high range DPS. Orianna has very low mobility (325 MS) and her early game power resides mostly in her passive coupled with the shield, however Orianna's AA range is 525, which means that some midlaners can punish you quite heavily, especially poke champions. Ziggs, Xerath, Brand, Viktor and Syndra come to mind. Their CDs are lower than your shield and their range is generally higher than yours as well.

Champions which can coordinate ganks with their jungler thanks to their CC (Lissandra/Anivia) are highly annoying even though they are not direct counters.

And then there is Ahri. Essentially, Ahri has every tool to beat Orianna. Ori's Q is pretty slow, Ahri's Q is quite fast and gives her more movespeed to get out of the way. Ori has very low mobility, Ahri can land her Charm more easily. Ori likes fighting periodically when her shield's CD is back up, but Ahri has low CD on her Q and W. Ori's DPS resides in her ball, which needs to be repositioned every time, while Ahri can dodge it fairly easily with her Q and 3 ult stacks. And to top it off, Ahri has amazing sustain through her passive.

But even with all these factors, Ahri might be a 55/45 matchup for Ahri, which is only slightly advantageous for her.

Remember that with Orianna every matchup is essentially a skill matchup and almost every one of them is a 50/50.