A few things to know about gyms:
Personal trainers are either terrible or expensive. If you can afford a personal trainer for a long time, either you have hired a great one because you are loaded, or you are paying for a crappy one and probably not getting great results. I know I am stereotyping, but I'm just trying to encourage a little skepticism. The gym I go to has multiple squat racks, rubber "bumper" plates for doing Olympic style barbell lifts, and has personal trainers/staff that have been stolen away from chains like Gold's, LA Fitness, etc. Why? Because the gym I go to has a premium business model where they pay their staff much more than other gyms do. Rather than paying a low monthly rate and then hiring a personal trainer, I pay a higher monthly fee for using my gym but don't need a trainer.
Anothe thing about trainers: Too often, their goal is to keep you coming back. They tend to have constantly changing, high variety workouts. These are fine, since the body adapts to exercise and the point is to prevent plateaus. However, this is not good if you are trying to learn to work out on your own. Also, variety is overrated for people trying to get fit. I've noticed trainers mix in strength with aerobic conditioning. Gaining muscle and losing fat are goals which are most easily met when they are done one at a time. If you want to get stronger, lift weights and eat a lot. You will make progress. If you want to lose fat, get up every morning and do cardio before eating ANY breakfast. Just water, then straight into cardio. Your body will be in fat burning mode with depleted blood sugar, and your energy expenditure will come primarily from fatty acids released by fat cells. But for the love of God, don't try to do both in the same workout. Cardio interferes with muscle building and vice versa, requiring ridiculously oprimized nutrition that is hard for people without professional chefs/dieticians to achieve.
Here is what I would look for in a gym:
1) 2 or more squat racks
2) 1 or more trainers who keep themselves fit using barbell exercises (my gym, despite having the standard cardio floorplan for the masses, has a section with the good strength equipment, and the staff who are into Olympic/Barbell lifting walk around and happily give advice/tips/form checks). When I am doing squats in the morning workout, I will frequently look over and see a staff member doing the same workout on the other squat rack.
3) Large numbers (5 or more) of benches for doing bench presses with barbells (not machines)
4) A rack of dumbbells that is complete (missing dumbbells is a sign of a gym that doesn't maintain their equipment) and ideally goes up to 120 lbs. per dumbell
5) Lots of pull-up bars