Stackish, 12 bytes (non-competing)
01d\+.qzcl2'
How it works:
0 Load 0 into stack (stack now 0).
1 Load 1 into stack (stack now 0,1).
d Duplicate last number of stack (stack now 0,1,1).
\ Swap bottom with top (stack now 1,0,1).
+ Add last two numbers (stack now 1,1).
. Pop to output (stack now 1).
q Undo pop (stack now 1,1).
z Pause.
c Clear screen.
l2' Jump to 2nd character (d).
d Duplicate last number of stack (stack now 1,1,1).
\ Swap bottom with top (stack now 1,1,1).
+ Add last two numbers (stack now 1,2).
. Pop to output (stack now 1).
q Undo pop (stack now 1,2).
z Pause.
c Clear screen.
l2' Jump to 2nd character (d)
...