Are push-ups, sit-ups, and squats (bodyweight) enough to build muscle?
First ditch the sit-ups. They can do more harm than good. Replace them with Planks.
Second add pull-ups. With push-ups and pull-ups you get the main push pull muscles activated. And you have many variations in hand positions, distance between hands and such to get every muscle working. Plus you can progress to raising your feet on the push-up to make it harder. If you get super strong and good balance, you can do hand-stand push-ups (sure admire anyone that can do those).
The Plank is fantastic for your whole core. Beats sit-ups hands down. And by increasing the time or elevating the legs or incorporating one hand lifts and such, it's infinitely progress-able. Plus can incorporate side planks once front planks are mastered.
And squats, another fantastic way to not only get your legs going but get some real cardio. Do them fast and you'll get your heart pumping at race pace. Your leg muscles are the largest in your body. To progress you can incorporate squat to jump which will enhance your fast twitch muscles.
There's only one piece of equipment you need and that's a pull-up bar. Get one with multiple hand positions and padded grips (especially for doing dead-hangs, for example). They run about $20. No other equipment needed for these four basic body movements.
(Go on the internet and you'll find guys with enviable physiques that they developed by doing and promoting nothing but body weight movements.)