One of the best glute exercises that you can do is the "Hip Thrust"... Because it activates the most "glute" muscle fibers.
You can do it at home or in the gym.
Anatomically you are in the best position to get the greatest amount of muscle contraction, and will have "constant" tension in the hips.
...And according to The Journal Of Strength & Conditioning, it's amazing for protecting your spine & knees from unwanted stress that is commonly known to exist with doing squats.
I like doing reps to failure or close, but I'm weird => start with 3-5 sets of 12 with 60-90 seconds rest in between.
Step #1 - Begin seated perpendicular next to a bench. Then lean back onto the bench so that the bench is near the top of your shoulder blades. Then with your feet firmly planted on the ground, push your hips vertically up, driving your hips up through your feet.
Step #2 - With your chin tucked to your chest, lift your hips up as far as possible. Then reverse the motion till your bum just touches the ground.
This is just the beginning.
Just adding "hip thrust" to develop your glutes would be amazing... but that's just one variation and modality for training activating and developing your glutes (i.e. single leg, double leg, inverted, weighted, rep amounts).
At home you can use: a couch, coffee table, medicine ball, heck even a kitchen table bench. Whatever stable piece of equipment that you have that's about that high off the ground.
Try it right now. Feels great. You can feel your bum fibers firing.