How can I determine the quality of training provided by a CFI at a local flying club?
Jeff, I really appreciate that you ask for safety before asking for the price. Unfortunately it takes some experience and time at a school to say if the equipment and teaching methods are safe or not, but I'll try my best to provide you with some of my insights into flight training.

I personally tend to the bigger flight schools. Having a strict training concept and standardized methods is a huge benefit in safety. If the same teachers also do more advanced training like IR or MEP your PPL training will most probably also be more intense. Your training should concentrate on learning to fly not flying the nicest routes but of course you should also enjoy your time. That's why you should of course also have a look at the people who are there. They should not only be friendly, but also be professional and well organised. The most annoying thing during flight training is bad organization and the changing of schedules due to absolutely avoidable reasons.

Good maintenance is key for safety. Here again, a big flight school that can do it's own maintenance has a lot of advantages. Often it happens that there are minor problems which should be fixed before the next flight without being a big safety hazard. It's better to squawk this and wait ten minutes until maintenance has fixed it, rather then flying without having it fixed, or cancelling the lesson. If the flight school has its own maintenance and they are happy with their work they will tell you and also show you around. Honestly the only thing you can do is to look if the hangar looks clean and tidy. If you decide for a smaller school without its own maintenance, it should at least be done at the same airport. Otherwise this could cause regular delays in your training.

Also think of yourself. During your solos and later when having your license you are allowed to and you will fly alone without an instructor. Be motivated to learn more than you have to. Know your aircraft and ask your questions. RTFM should not be the general answer of your flight school but you should spend some time doing it.

I made a list with all my questions before visiting different flight school so I don't forget to ask them. I would sugest you to do the same before getting there. If you for example ask for some more detailed information how many flight time you have to do with an instructor before flying solo and what exercises and maybe progress checks you'll have to do prior to it, you'll get a good insight how structured the flight training is. Also don't forget to ask about the theoretical part. That's really dependent on your self if you prefer to learn with books or with a teacher in front of you.