Unless your phone has a USB On-The-Go port, then no you cannot plug it direct into the Arduino. It is highly unlikely that you have an On-The-Go port, or if you do that the software exists for it to enumerate an Arduino. And no, you definitely can NOT directly connect the TX and RX lines of the Arduino to a USB port.
Yes, you can communicate with bluetooth if you add a bluetooth UART adapter to the Arduino (HC-05 / HC-06 are common). There are many apps for mobiles (both Android and iOS) for controlling Arduinos through bluetooth.