This answer will be a HUGE generalization, because I improvise/adapt to what someone needs on the spot based on their goals, their skill level, their level of nervousness, their experience level, age, etc. In addition, I sometimes get really specific requests like "I don't want to learn how to sing but I want the benefits of easier breathing". Stuff like that. So keep that in mind as I describe how a lesson will go.
Typically, I'll ask you about your history with singing, your interests, goals, musical tastes, etc. I'll listen to your speaking voice on one or more vowels, connect that to a pitch on the piano and have you sing (depending on whether you know how to match pitch or not) up and down some scales. Then we'll map your Vocal Comfort Zone.
At that point, if you have a song that you've sung recently and if you feel comfortable enough to sing it (first time singers are sometimes way too terrified...that's completely okay, we'll get there as you feel comfortable/ready in future lessons) then I'll have a listen.
From there, we'll get to know your vocal habits with singing and possible exercises (depending how much time we have and what questions you might have) to either clearly sense how you're using your body when you sing and/or how to do something different that feels better and move more towards the sounds you're looking for.