No, there are no particular prerequisites for indoor climbing. Strength, flexibility, endurance will come along the way. Just take a look into any gym, you will see small, large, fat, weak people. In my place there regularly is a one-armed guy...
Look for a large, well-run, popular gym. Good signs are:
- A nice, inviting website which shows all rooms/walls of the gym; explains how everything functions; explains any rules.
- Beginner and advanced courses.
- Being chock full at prime time.
- There should be a small gym-inside-the-gym with assorted equipment (boards to hang from, maybe some weight training stuff).
I would strongly suggest taking a beginners course. Those are available for adults. While it may feel weird, just shrug off the feeling. Do not be shy, talk to the coach, ask questions, actually do what she says, and so on.
But the main reason for taking a beginners course would be to find another beginner to start your journey with! Exchange phone numbers with someone who seems to be close to your level of body strength, fitness, etc.. Find a weekly "jour fixe" where you regularly meet. The time/day-of-the-week of the course is a good starting point, as both of you obviously had spare time right then!
Then just go and climb. You will find that you will improve very quickly. Do not be shy watching better climbers (i.e., everybody else). Nobody will think bad about you; most people will remember how it was when they started.
The partner is important because a) motivation and b) talking about how to tackle different problems. As both of you will have different arm/leg lengths, mobilities, etc., you will quickly see that you will take different approaches to climbing, so you will learn from each other, even if you both are beginners.
I tried to get into indoor top-rope climbing twice, and it did not work out. The problem is that you are absolutely tied (sic) to having a partner around (so no spontaneous night at the gym) and downtime is much worse. One of you is always not climbing; and when the gym is full, the fullness is much worse, so to speak, since each climb takes longer. Also, frankly, I had incidents with lazy/inattentive partners while my life was literally hanging from a thread...
Bouldering to the rescue! I encourage you to skip all the complexities of top-rope climbing, and go straight to bouldering, if you have a gym that focuses on that. Bouldering might not look like much, but it is very intense, and since you do not have a long climb for every route, it is much easier for the route setters to give you many, many more diverse problems.
If a bouldering hall is full, it's not such a big problem. You get queues in front of the walls, but since each attempt takes only a few dozen seconds, 1-2 minutes max, you get to climb rather sooner than later. Also, when the difficulty rises, you need the time between tries to recover your strength anyways, so it is actually good to see what others are doing at "your" problem.