how do you make quick moves?


Not sure what might cause issues on a tablet, but enabling premoves is important for speed games, which you can do here:
https://www.chess.com/settings/live
A premove is when you move during your opponent's turn and chess.com will remember your choice. After your opponent moves it will be played instantly for you (you lose 0.1 seconds off your clock) as long as it's still legal.