Push-ups are usually considered an upper body exercise that develops the chest, shoulders, triceps, and upper back. But when done correctly, they are a very much a core exercise, too.
The top of a push-up is a "pedestal" plank which is a variation of the more popular plank done on the forearms. This makes a well-done push-up an excellent way to work the core.
Once you have your push-ups dialed in, you might want to make them harder by using a resistance band. Proper band placement can enhance the core stabilization effort as well. See more in the video.
Be careful not to start using the bands too soon. Make sure your push-ups are excellent first...otherwise The Form Cop might catch you!
Be seeing you.
-gary