Rule #1

The specific exercises you choose within this realm are not the concern. What is important is that whatever movement you decide to utilize, make sure it answers "yes" to the question: "Hypothetically, if I could remove a significant amount of body fat right now, would it immediately make this exercise easier?" If so, master that exercise.

Good Examples: walking, push-ups, pull-ups, jump squats, practically any TRX exercise, sprints, hiking

Bad Examples: deadlifts, barbell squats, sled pushes, power cleans, curls

These "Bad Examples" are NOT bad exercises and in fact, they can all be a valuable part of any training regimen. However, a reduction in body fat would be unlikely to improve the performance of them. 

Your body is smart. It knows how to respond to your environment and change in favor of your survival. If you consistently perform exercises that depend on your ability to manipulate your body weight, you will mold yourself into the best framework for doing so. A body capable of moving itself tends to be lean, muscular, balanced and proportionate. 

You do not need to drop your current exercise program and replace it with these exercises. However, if weight loss is your main priority, make sure to ADD THEM into your current routine. 

You do not need a gym. Do what you can, when you can, with the resources you have.