Switching between pins upon a click of a button is fairly easy. You need a loop that's reading the button state constantly and upon change (with proper debouncing), you change the digital pin that's high. You can use a state counter to figure out which pin is being lit.
Where I have a question is the ten seconds that you want the pin to be high (LED lit). What happens after ten seconds? What if you don't press the button anymore? Does the LED simply turn off? And what happens when you press the button again? Will the sequence start over? Will it simply continue where you left off?