What kind of results are you looking for? You can only go so far without lifting.
Have you considered buying equipment to train with at home? You can also improvise equipment using heavy bags for resistance with for example push-ups, pull-ups, squats, lunges etc. Or you can find heavy rocks and lift them.
The way to gain muscle while burning fat is to eat enough protein and other micro nutrients for muscle growth, but keep the calories 200-300 below your daily requirement.
I'm pretty sure Bruce Lee found a way to gain muscle without hitting the gym. He was shredded/ripped. Although I don't know this workout but I think a simple Google search will suffice. but if you plan to work with your own body weight you'll only get so far.
If you want to gain muscle and lose weight but don't want to lift weights then you're gonna have a bad time.
I suppose you can start running (lose weight) and push ups and crunches will help but you would have much better, quicker, results if you started lifting and went to a gym.
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Resistance bands and body weight exercises.
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