Push-ups are good - they're one of the most effective all-over moves you can do with your own bodyweight. To bulk up for the film 'Warrior', Tom Hardy focussed solely on push ups, something crazy like 800 a day, and in that movie he is RIPPED.
A pull-up bar can be really effective too. You can buy ones from Amazon for pennies that hang from doorways via a mix of gravity and momentum (so no messy holes needed to hang them up). They're great for building upper body strength, developing the large-shoulders-thin-waist T shape that women love. They can be used for a whole bunch of different exercises too. Before I decided to transition, I wouldnt have a training regime per se, but every time I walked past the bar I would do a few quick sets.
Though the best, most verstatile gym object I'd recommend if you
only had to get one would be a set of dumbbells. They can be used for so many different moves. Variety improves your body's capacity to learn, its muscle building, and will also keep you motivated and interested. They can be stored anywhere (I keep a set in the garage which I use from time to time to banish the saggy-ness!

). Like the pull-up bar, they can be purchased for buttons too.
As Kenton and Aleon said, protein is your friend! It basically fuels your muscles, helps them repair and increase. Though simply adding more into your diet alone won't do anything, you need to keep up the exercises but should notice a difference.
Hope this helps!

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