After looking into lerp and playing with its parameters for a while I believe that you can get all the results that you need just from it.
For example for ease-out you use:
lerp(start, finish, time_counter ^ 2)
For ease-in you use:
lerp(start, finish, time_counter ^ 0.5)
You just need to make sure that:
1) your time_counter starts in 0
2) you increment it by some amount depending on how fast you want the movement to be (0.1 each tick its a fairly fast movement, so I normally use 0.05)
3) your end condition is always time_counter = 1 (or greater).
Finally, if you you you are looking for a higher-level solution you could always use the tween plugin as already suggested.
I really like the art style of your game. Hope this was helpful :)