I'm not the OP, but I seem to be getting good results (strength gain and fat loss) from a combination of dieting, beginner calisthenics and a bit of weight lifting on the side (dumbbells / kettlebells at home).
If you stay in a calorie deficit and eat enough protein, you can lose fat and gain muscle at the same time which is nice :)
Cardio seems to be good for overall health so I do a bit of that too sometimes, but I don't think it's strictly needed for fat loss.
Some form of exercise and a sustainable diet is all that's needed really. If you do both of those things in any way, you'll be way healthier than the average person.
