I am a gold ranking user, and don't know how to stay in the teamfight phase with Orianna.
Orianna is a control mage, which broadly means her primary focus in a teamfight is to stay at around mid-range and utilize her spells to deter enemies from forcing an engage and apply crowd-control spells to help her team secure kills.
For Orianna specifically, though, her primary form of control is a combination of her Q -> W, and her Q -> R. Her Q -> W is a simple combo that does a small amount of chip damage early, but can scale up significantly as the game progresses, often times dealing ~50% of an enemy carry's health in damage. When an opportunity to apply this poke damage presents itself, you should jump at the opportunity. If the enemy charges forward in an attempt to deal some damage back to you (otherwise known as "trading") you should hold the Alt key and press E to withdraw the ball back to yourself to apply a shield. You can also use your E on allies who are getting engaged upon to mitigate the damage they take.
Your most explosive ability is your R. In a somewhat large area around your ball, you pick up and throw all enemy champions, majorly disrupting their positioning in a teamfight and dealing a large amount of damage. This spell wins games. If you see an opportunity to use your Q -> R (or use your E on an ally that's surrounded and press R, or use your E on an ally that's about to dive into multiple enemies and press R), you should absolutely do so. However, it's important to note that you shouldn't hold onto your ultimate for too long. Some players have a hard time balancing knowing when to hold their ultimate and when to use it. It all depends upon the fight, and oftentimes using it on a single enemy carry alone can win you a teamfight. In other words, don't greed for a high-target ultimate when a one-target ultimate on a high-value target is available.
In summary, Orianna should be played as a mid-range support and control mage at the teamfight phase. As a squishier mage, you should be monitoring the enemy divers and assassins and ensuring that you're safe from their engage tools (jumps, dashes, ultimates) and applying your poke damage where it's safe for you to do so. Shield your allies from chip and engage damage, and make sure you can get off a high-value ultimate cast, ideally when following up on a tankier ally's engagement, but mostly when an opportunity presents itself.