There's a few things you can do, but these things have some limitations..........
1. Pushups with extra weight added
You can take a backpack and add weighted objects to it (books, etc), then do pushups with your body weight + this additional weight. It's a short term work around for not having access to weights and you'll be able to make some progression beyond doing bodyweight only.
2. Pushups as per above at various angles to work upper chest
When you do pushups, put your feet up on a higher object (i.e., a chairseat) so that your body is angled downward. If you hit this right, you can activate your upper chest a little better
3. If you live in a big city, most have outdoor parks with a track. Many of these have pullup and dip bars. If so you can do dips there for chest and triceps. When you want to add weight to this you can get a dip belt and a cheap DB from Sports Authority, Big5, or whatever store in your area. Or if you don't have the $ for that, a dog leash and some creativity can work just as well. Look up community centers in your area. Use Yelp, and search for reviews that include tracks. Worst case scenario, find a elementary school playground and get creative.
4. You can also do dips at home on a bathroom counter top if it's stable enough. You have to get creative with body positioning and this all boils down to safety. Don't be stupid with it because it's not worth it.
5. You can also get creative with some cheap bands from Sports Authority, Amazon, or even Vitamin Shoppe and do Chest Flys off the floor. I don't like these but I'm throwing this out there because you may try them and like them.
Bottom line is just get creative, there are ways.....plenty of dudes with nothing figure things out. I did the above things when I was a kid and didn't have access to stuff. They can work beyond bodyweight only, but up to a point. Weights/gym is best though. There are also YMCAs and community centers that may have access to these things for free so unless you live in a very rural area, you can check those out as well.