I'm used to riding a single speed bike and only once in a grand while did my chain drop.

I've started riding a multi geared bike with an Ultegra drive train (rear/front derailleur, cranks, shifters) and I've dropped my chain a few times. It's a double ring setup and I'm always in the smaller ring.

I'm new to riding gears so excuse me if I failed to add any more detail, or if I'm asking a stupid question. It dropped during a cross race today on a bumpy descent.. I have a feeling I just need to pick better lines, but is their something else to this?

There are 2 limit screws on the front derailleur that prevent it from moving too close and too far from the frame. The one that prevents it from moving too close (the one closest to the seattube) likely needs to be tightened a bit. (Turn clockwise 1/4 to 1/2 a turn at a time.) After adjusting, take it for a spin to make sure the chain doesn't rub in the big-cog/small-chainring gear and that it still shifts effectively between the big and small rings.
For cross bikes, I'd recommend one of those plastic chain-watcher things that clamp around the seattube under the derailleur. They keep your chain from hopping off on the inside:
https://www.rei.com/product/670913

+1
Either the third eye chain watcher like the one in the link, or a deda dog fang (you'll have to get the deda dog fang that fits your seat tube diameter if you go that route).

Use one of these devices, and the problem will be solved-