The first thing you should do, regardless of where you've decided to meet, is to check the teacher's photo ID, CRB check, and any other supporting documentation they claim to have. Don't worry, they won't take offence at this; they know to expect this as professionals. It is best to do this as soon as you have met and introduced yourselves, so you can concentrate on what really matters: playing music!
We always recommend learning in a quiet room if possible so you can both work without disturbance. This said, we would suggest as good practice that the door be left ajar and that parent or guardian pop in to see how things are going. This is reassuring for all concerned. A parent or guardian should always be present in the home during a music lesson.
If you have arranged to go to the music teacher's home, please think safety first and ensure that the tutee does not visit alone until comfortable with the tutor.
It is helpful to consider exactly what you want to achieve from your music lessons, and to explain this to the teacher. The teacher may want to talk through your motivation for learning an instrument, and may do some exercises with you to ascertain your skill level so far - don't be bashful, you're not supposed to be a maestro at the instrument yet!
Ultimately, the first lesson is an opportunity to gauge your rapport with the teacher and their teaching style. Don't be afraid to ask your teacher to raise or ease the pace or difficulty of lessons; equally, if you would like to change anything about the lessons (direction, structure, etc) just mention it to your teacher.