Most people go to a professional stylist or barber when they want a haircut. You describe this service as:

I got my hair cut.

Sometimes people will also say:

I cut my hair.

But that's a little confusing because it sounds like you did it yourself.

Aside from getting your hair cut, other things you can "get done" include:

• get your oil changed (on your car)
• get your nails done
• get your eyes checked (by an optometrist)