Push-ups and chest presses do work the anterior shoulder and triceps as well, as synergists (helping muscles). If you want to feel these more in the chest, you can try pre-exhausting the chest muscle by doing a set of chest flyes (an isolation move) right before doing the push-ups/presses.
Also, you can experiment with hand placement when doing push-ups. Sometimes just a little modification in hand width or how far forward or back the hands are can put more emphasis on the chest. Also, think of keeping your chest out and relaxing the shoulders back and down as much as you can.