(1) Do the rinks in your area offer public sessions during weekday mornings? If so, does your schedule permit you to skate during those times? Taking private lessons during crowded public sessions generally is a waste of time and money. Some rinks though do cone off a section of ice for private lessons during otherwise crowded public sessions. On the other hand, taking beginner private lessons during freestyle sessions can be intimidating (e.g., dealing with jumpers, and observing protocols for right-of-way for program runs). Some rinks have designated low and high freestyle sessions; but others don't. If possible, weekday morning public sessions are the way to go: typically not crowded as long as schools are open, and a lot less expensive than freestyle sessions.
(2) When starting out, it's perfectly OK to try out a lesson or two with different coaches to see which ones you like. Just tell them in advance what you're up to. And once you've picked a coach, and if after a while it becomes clear that things aren't working out, don't be afraid to end the lessons and move on to another coach.
(3) One place to start is to ask the figure skating directors at your rinks for recommendations.
(4) But I agree with WednesdayMarch. The best approach is to ask other adult skaters and to observe coaches and adult skaters in action (if you can). Follow proper etiquette: don't interrupt or otherwise interfere with a lesson (e.g., get in the way); and don't freeload (e.g., if the coach is demonstrating back X-overs, don't start practicing back X-overs yourself).
(5) It's important to find a coach whose style matches your individual needs, which will differ from those of other adult skaters. E.g., some coaches do little skating themselves during a lesson, spending more time on verbal instructions; others use a follow-the-leader style, with little explanation; yet others are more analytical. You also need a coach who understands your overall physical condition. Several coaches I know have a "Just do it!" attitude. Maybe great for whipping younger skaters to perform better faster; but for older skaters that are not in great condition, doesn't work too well, and somewhat dangerous in my book.
My coach shows me how a move is supposed to be done. Then analyzes what I do wrong (you're dropping your left shoulder; you're leaning too forward; ...). If it becomes clear I can't make further progress in my current state, she cooks up a regimen of remedial drills for me. She's perfect for me. I've found my heel-and-sole mate.