Grant's answer contains a lot of great information/tips about improving your push up workouts, but I would like to add a few more:
Dips
Ensure your body is leaning forward during the exercise, and that you completely lock out at the top (not just your arms but your shoulders) and focus on using your chest for this. This will exercise your pectoralis minor which is under your pectoralis major, this muscle won't add much strength but just a small amount of size increase on it will push out your pectoralis major and make a giant impact on chest size. Mind-muscle connection really does make a huge difference when building muscle.
Pec flies
This one may be unreasonable depending on your equipment but you can get some exercise rings or a trx setup relatively cheap and have the ability to do pec flies with virtually as much (body-weight) resistance as you will ever need. Pec flies are huge for building chest size as they are one of the only exercises you can do to stretch your pec through it's full range of motion, and that is key here, bring your arms back as far as you possibly can (don't be afraid to drop the resistance by increasing your incline). This exercise is all about the range of motion and not so much about the resistance. Again mind-muscle connection is key here. Additionally I know you asked for body weight exercises but you can also get some fairly inexpensive resistance bands to do these with that can also easily be done in your own home.
Explosive push up
This is covered by Grant's clapping pushup but the alternative to that that I use is to put some stacks of plates on the outside of my hands (can be substituted with boxes) and "exploding" off the ground high enough to catch myself on them. This type of training will really transfer to the bench press in a big way if you ever start weight training.
Isometric Chest Squeeze
Stand upright with both arms out in front of you, bent at a 90 degree angle. Lock your hands together and squeeze as hard as you can. Hold the contraction for 15-30 seconds then relax. Isometric exercises aren't great for increasing strength but I like them because you can basically get out of them what you put in, and no matter how strong you get this one will increase in difficulty with you as one side of your body will be able to resist the other with relatively equal strength.
I mentioned it a few times above but I want to re-stress the importance of mind-muscle. It's important to take a few minutes to really learn your anatomy, and how/when the muscles you are targeting are fired. You can do pushups using your shoulders and triceps almost exclusively and not really get a good chest workout. However if you understand how the chest comes into things and focus on using it, it will make a larger difference than almost anything else you could do.