I have gotten lots and lots of students over the 20+ years I've taught. The first lesson is SO important.
Things to remember:
If the kid/adult (let's go with "kid" for simplicity's sake) has never taken music lessons ever before, that's pretty easy. They're excited, nervous, and incredibly unsure. But a lot of reassurance usually settles them in.
If they have taken flute lessons before, you are being held up to the memory of the previous teacher for quite a while. Even if they left the previous teacher on bad terms, you don't want to ever EVER talk them down. Don't encourage it, don't do it. Use lots of "well, we all have different approaches" or "sometimes, no matter how good the teacher or student, the chemistry isn't there, and it happens to everybody at some point."
There are plenty of ways to prove that you're the superior teacher/player/person. Being a bitch is NOT one of them. And even if you don't really know the person, say, "Of course I know her! I think I met her at... "fill in the blank with some big concert or class or something). "What a gal!" General statements can cover a multitude of non-comments.
Ask the student what the other teacher did that was wonderful. Be admiring. Be respectful. Praise the student for learning so much. Say that you hope to continue the techniques that the previous teacher did (the ones that worked), but in your own way. What were the favorite things in each lesson? What about recitals and such? Remember these, add to your repertoire. Make a note.
Then, tactfully, ask what didn't seem to work for that particular student. What wasn't too fun? (Do NOT give in to the temptation to be snarky or know-it-all. You can NEVER take it back) What did you do too much? Scales? Memorization? Band music? What did you wish you did more of? Sight reading? duets? Again, make a note.
Remember: even if the student didn't take FLUTE lessons before, they may have taken piano or violin or voice lessons. These can greatly impact the baggage the student brings to your lessons. Ask. "Ever take other private music lessons? On what instrument? What was that like?" If you yourself ever took piano/etc. lessons, share that. Tell a story or two about your old teacher. REMEMBER to be positive, funny, supportive. Not dire and dreadful. You can joke about how you hated a particular piece or exercise. But again, don't go negative. You want the assumption to be that this will be GREAT. If they talk about how much they hated a teacher or how scared they were, be sympathetic -- but just for a moment. Then move on to how GREAT it will be with the right teacher. Suggest that lesson teachers have come a long way since "old school" teachers.
If the parent is there, still address almost all of the discussion and your attention to the student. Don't ignore the parent, but you'll be a better teacher if you maintain your focus. Towards the end of the lesson, give a brief explanation to the parent about what you expect of the kid for the week, and how to help them. Be sure to compliment the parent on SOMETHING about the kid. Don't get all weird and ridiculous (this kid will be in the New York Phil by the time she's 18!), but a casual comment about how well-behaved, cheerful, fun, articulate, or attentive the kid was can go a long way toward a good relationship. Don't lie, though. If you can't find a single genuine thing to compliment, perhaps the thing you should be saying is, "You know, I think lessons will be a great success after she's finished the school year, and has time to focus," or "While my openings right now are mostly for advanced (or older/younger) students, I know a teacher who is accepting younger/older/beginner students right now. She and I work together -- she sends me advanced students, and I send her wonderful beginners when we don't have room." etc.
An aside: try to get the student playing as fast as you can at the beginning of the lesson. Don't wait too long or they'll get all nervous. Suggest they warm up on something while you rummage around in your bag, looking for handouts or a book or something. Reassure them that you're not listening. I usually sing to myself or something to make noise. This really helps break the ice.
Then? Ask them to play "that nice song you were just playing" again. BE SURE to SMILE. I don't care if they're playing 3 Blind Mice in 3/4 and in minor. SMILE. And be sure that they AND THE PARENT see you smiling. Be subtle.
Then IMMEDIATELY, when they are done, tell them what are the good things you see. For example:
"Wow! You've got SUCH good posture. And I sense that you really like this song?" Smile.
"I'm hearing good tonguing, such a nice strong low sound, and you really have a handle on your fingerings!"
Do NOT say the negatives. Do NOT DO IT. Just stop there. Smile
Now, the most IMPORTANT thing you can do. Really. Please pay attention.
Ask the kid to tell you THREE GOOD THINGS about their playing. This is hard because they always want to tell you what's bad. Interrupt them, and say, "No! Good things only!"
I usually give examples, so they don't stall out.
"What's good? some possibilities: good tone, good high notes, low notes, tonguing, playing fast, playing slow, vibrato, sightreading, posture, hand position, and so on?"
Now here's the important part. These THREE THINGS are things they are proud of. DO . NOT. criticize these things in the first few lessons. I don't care how bad their high notes are. If they say they're good, just don't say a darned thing today about this. They have just shared the things that they hold dearest about themselves. They don't know it but they did. Be smart.
When you DO eventually hit these topics, here's what you say. "Now, your high notes are off to a REALLY good start! We're going to build on this strength! What can you do, you think, to make them even better?"
(It's the "even" that saves you from being evil.)
After you've asked about the THREE POSITIVE things, you can ask them "Now, what are three things you want to improve?"
Do NOT say "three things you're bad at." They are now YOUR student. THey're not bad at ANYTHING. They are awesome, and they are simply trying to bring up ALL their skills to match their strongest ones. Period.
Then? If they start rattling off tons of problem areas, interrupt them. Say, ONLY THREE! And laugh. And remind them that the things that they have to work hardest on will eventually be the things they are strongest on. Tell an example of what YOU'Ve worked out. I tell them how hard I worked on tone (for at least 3 years in college), and then I play a little bit of something luscious, and say, "See? I now have a rockin' good tone, and I know how to TEACH it. But if I had started out naturally good at that? It would be harder for me to work with my students."
Finally, be sure to PLAY for them. Little bits, here and there. Memory, please. Just a minute here or there of something you're awesome at. And then, don't treat it like a performance. You're just casually giving an example of something you are discussing (like tonguing techniques, vibrato, tone, high notes, whatever). Let them be impressed, but DO NOT appear to be impressed with yourself.
Follow up the lesson with an email or phone call to see if what you discussed in the first lesson is still making sense. Make it super quick, and cheerful.