That is a regular Mega2560 Shield. You just plug it in and go.
Install the TFT_HX8357 library from Bodmer. i.e. https://github.com/Bodmer/TFT_HX8357
Edit the User_Setup.h file to select ILI9486
Run all the library examples.
I don't think it is available through the IDE Library Manager. You will have to install from a ZIP.
You can access the unused pins with a short piece of angle header strip. Connect your push-button between an unused GPIO pin and GND.
David.