I'm going to go to the gym for the first time next week.

It's a small gym in a small village (not even a town) so maybe the coaches are not competent.

My goal : I want to lose fat and gain muscles. But I don't want to have big muscles while not being strong. Because I don't like how this kind of muscle looks. You can tell it's not dense muscles.

I want muscles who looks strong, powerful, even if it means they will be smaller than they could be otherwise. I also want to build a muscle base that will remain as much as possible when I quit going to the gym (I plan on going for a few years and then stop to go back to martial arts in the best shape I can). Durint this time I want to put on muscles that I'm more likely to keep even if I quit going to the gym. I don't know if it's possible.

Lastly, I am recovering from an injury. Last step of the re education is to strenghten my body which is very weak. i have weak joints etc.

What should I tell the gym guy in order to get a program that will suit my needs? (they design a program for you) and how can I know if he is good or not? Finally, what advices would you give me regarding my goals?

Tl;dr: my goals are strenghtening the body, getting a muscle base that will stay for life and doing so with security because my body is weak and fragile