I will not give you code, only clues on how to write the code yourself.
The way to tackle this is to (at first) remove the Arduino from the equation. Replace it, instead, with you.
Imagine you are doing the timing yourself. How would you go about doing it?
This is how I would set about the task:
- When the first button gets pressed I would look at the clock and jot down the current time.
- When the second button gets pressed I would again look at the clock, and subtract the first time from the current time to find out how much time has elapsed. I would jot that down.
- When the third button is pressed I would do the same again - look at the clock, subtract the start time, and jot down the difference.
- And I would keep going until I had run out of buttons.
So now you need to find the right code to read what buttons are doing, get the current time, and finally send the times somewhere useful.
For all those there are many many many examples on the web.