Having no recent experience I don't think I have the right vocabulary, and am very interested in playing classical. Should I make sure they have some sort of curriculum material, like Alfred's, as well?
If classical music is your prime interest, it behoves you to learn and master all the basics properly
- that means note reading, rhythm (note values, beat counting) and developing both hands equally well. What your RH learns, your LH must also learn equally - I cannot emphasize how important that is for classical music.
Which is why I don't approve of Alfred's and other method books of its ilk, whose aim is to get adults playing recognizable tunes and harmonies right from the start, based on RH melody and LH chords (otherwise they'd be bored and give up, right?), at the risk of lop-sided hand development and giving short shrift to essential basics. (Children's beginner primers tend to be much better.........). Ideally, your teacher is used to teaching serious adult students who want to learn everything
properly (i.e. don't want to just learn what they want to play - Moonlight, Für Elise.....). Unfortunately, many - probably most - teachers don't want to teach adults, and those who specialize in adults often just teach them what they want to learn, no more and no less.....
I'd look for a teacher who teaches along traditional (classical) methods and depending on where you live, most teachers will fit your bill or hardly anyone. In the UK (and quite a number of other countries around the world), almost all teachers teach that way, because almost all students do ABRSM exams, which is about as traditional and classical-orientated as you can get. This is the ABRSM syllabus for piano (each grade takes on average a year): https://us.abrsm.org/fileadmin/user_upload/PDFs/Piano_Syllabus_2019___2020_complete.pdf
If you're in North America, you might prefer to look at the RCM syllabus; in Australia, the AMEB.
In matters of qualifications, in the UK, almost every teacher has a teaching diploma or equivalent. Not so in the US, but remember that it's no use having a conservatory-trained concert pianist as your teacher if he doesn't know how to teach. And don't rely on what you see in 'student recitals', for obvious reasons. If you can get recommendations from other students who have the same goals as you, that might be your best bet.
Ask your prospective teacher if they will ensure you learn all the basics based on classical music (not pop or jazz) - and preferably, follow one of the syllabi mentioned above. (You don't need to do the exams, but I'd recommend them). The good thing is that once you have grasped the basics, there is no lack of original piano/keyboard music (for all skill levels) by great composers to learn from, without having to resort to rubbishy arrangements.
But don't forget - learning is a two-way street. You need a good teacher but you also need to be a good student, and practice daily and purposefully and with intent. And you need to be very patient, which is where many adults fall on. Progress will almost certainly seem very slow, but slow and steady always wins the race when it comes to mastering the piano. Your brain will certainly skip ahead of your hands (whereas with kids, it's often the opposite) but what you think you can do isn't the same as what you can actually do, so be guided by your teacher and don't be tempted to push too fast, just because you think you can. Teachers may let adult students push them in directions they wouldn't allow child students to do.....