How do I clear out the bathroom drain?

First of all try using a plunger. Cover the overflow hole with wet paper towels. Pour some baking soda down the drain. Then pour some vinegar down. After it stops foaming follow up with hot water. If this doesn't work the clog may be further down the line and it would need to be rodded from underneath.

Here is a method you can use that works well.

https://www.bhg.com/homekeeping/cleaning-and-care/cleaning-advice/how-to-clean-bathroom-sink-drain/

They make plastic little "snakes" of sort to pull out hair from your drain. You didn't mention if this was the bathroom sink or the tub but should help both. I know it did for me. Got mine at amazon.com and paid less then $8 for a package of 5 and they are reusable.

Hello. Even though we have an auger, most commonly we find that our drains need to be physically fished out. Either using an unbent hanger as a refashioned hook or plastic barbed tool from the dollar store to explore the drain obstruction.

Fishing out the blockage with that in -our experience— to partially involve hair seems to most readily alleviate stoppage in our bathroom sink and shower drains.

https://www.hometalk.com/diy/repair/plumbing/ways-to-unclog-a-kitchen-sink-drain-44244555

Ordered the plastic drain cleaners and will give the baking soda and vinegar a try! Thanks for your help!