Well, that all depends on who you are and what you want.
A child, for example an 8 year old beginner, would be taught in a traditional method progressing towards the classics. During the first lesson your child will be taught how to play a song or two even though they wouldn’t yet be able to read music. Note reading usually happens in a few weeks, a little at a time. They will also be taught music theory and piano technique. When the student has enough faculty, they can play whatever music they choose. In addition to the traditional method, students will learn some improvisation as well.
For a child of any age who’s previously had piano lessons, their first lesson is different. I will ask them to bring their books and play anything they choose so I know how to proceed. They will continue to learn technique, music theory, and improvisation in the traditional method.
In the case of the older beginner, maybe a teenager or adult, they will be taught a song or two at the first lesson, but your piano books would be appropriate for your age group. You would also be taught music theory, technique, and optional improvisation.
Adults and Seniors
When it comes to older adults who have played before, it has been my experience they they usually want to play what they want and most people don’t want to be bothered learning or relearning technique, so I usually alter their pieces so they are easier to play and more enjoyable for the student.
I hold student recitals. It is a very good experience and gives the student something to work towards. Most of my students are happy to participate, but I don’t require it.
All my students learn to accompany either themselves or someone else. Many times it’s for accompanying their school choir.
There are some people who do not want to learn improvisation and I never force the issue. There are others who are accomplished, but only want jazz or improvisation lessons. I try to meet everyone’s needs.