Where are your contacts? Are they ones you've saved to the device (SIM card), or ones you've saved to Google or Samsung contacts?
I have two Contact apps listed in my app drawer. One is the Google app, the other the Samsung app. In the Google app when I open the contact there is a blue circle that appears at the bottom of the screen for editing. Bottom right corner and if I scroll it actually says the word "Edit contact".
With the Samsung app. At the bottom of the screen three tabs appear, Generate QR code, Edit, Share.
Since the formfactor of the S7 is smaller they may have hidden these options just off screen to provide maximum viewing for the contact information, so try scrolling up and down to see if it reveals its self.