Buttons for a timing game
Connect both buttons to the same ATmega328 port (e.g. PORTB), then read the status of all 8 of the PORTB pins simultaneously by reading PINB, the PORTB Port Input Pins register. See ATmega328 datasheet section 13. I/O-Ports, section 13.4.4 PINB – The Port B Input Pins Address. Assuming you're just getting started with programming, this is very similar to what you've been doing. The arduino's digitalRead function actually reads the selected Port Input Pins register, but then it only tells you about one of the eight pins. It's actually simpler to just read the PINB register directly, since you care about reading all 8 input pins simultaneously. Be sure to test for the case where both buttons are pressed simultaneously.

Interrupt handling is another approach, but since you indicate you're concerned about determining which button was pressed first, be careful to understand how interrupt priority works. In the event of a "tie" between two interrupt sources, the higher priority interrupt wins.