“I need to go get my prescription filled, so I'll be back in a while.”
You had a pain in your shoulder. The doctor prescribed pain killers for you. You told your roommate about this, but now you need to go to a pharmacy to get your medicine. You tell your roommate this.
I need to go get my prescription filled, so I'll be back in a while.
When you "get a prescription filled", you go to a pharmacy and the pharmacist gives you the medicine that is written on the prescription.
This is what you say when you're leaving somewhere, and you want to tell the people there when you're going to return. You don't need to say this every time you leave your home or office, only when it's important for people to know what time you're coming back.
This is a way of saying when something is going to happen. It's not very specific. It usually means some time later in the same day.
In casual speech, you say "go (do something)" to mean "go (somewhere) and (do something)". For example:
Do you want to go see a movie?
This means to go to a movie theater and watch a movie.
Could you go find me a screwdriver somewhere?
This means to go to the place where tools are kept and find a screwdriver for the speaker to use.