These are the basic pieces of gym equipment you need for a small fitness studio or in-home gym:
A training bench for weightlifting
A dumbbell set with multiple weights
A barbell set for weightlifting
A kettlebell set to activate different muscle groups
A pull-up bar (and frame)
A treadmill, rowing machine and stationary bike for cardio
A fitness ball for a variety of different exercises
Accessories like a rollout wheel and stretch bands for aerobics