You have some nice art there! That man's running animation is hilarious!
To make something acellerate.
Basically, in your movement code, you are currently saying
set x to self.x + 40 * dt
or something, right?
To add acelleration, just change the constant to a variable:
set x to self.x + speed * dt
and every tick, or every X seconds, add something to that speed variable.
If you don't understand, mention me with sqiddster so I don't lose the thread ;)