Run a loop-back test...
Disconnect power from the board
Remove all connections and shields from the board
Force the processor to remain in reset by connecting a jumper from RESET to GND
Connect a jumper from the TX pin (1 on the Uno) to the RX pin (0 on the Uno)
Connect the board to your computer. After a brief pause Windows will produce a device-insertion tone if sound is enabled. Linux may or may not produce a device-insertion tone; an entry is added to the system log.
Start your favourite terminal application. Serial Monitor will work fine.
Connect the terminal application to the serial port for your board. The baud rate is irrelevant.
Send data by typing. Everything you type should be echoed back. To send data, some terminal applications, like Serial Monitor, require pressing the Enter key or clicking a Send button. If exactly what you send is echoed back then the board passes the test. This means that the host computer hardware driver, USB cable, and USB to serial converter are all working.
Shutdown the terminal application
Disconnect the board from the computer
Remove the two jumpers