That sounds like an interesting project! What are you going to use it for? Pardon my suspicious nature, but it sounds so very much like an exercise given in school/college. I dont do other persons homework.
If it is because it is a nice challenge to programming, then I suggest you start by writing a sketch that can read a button and send a single character to Serial. What you'll find is the first minor hurdle, "switch debounce". When you fix that you can expand the program to count the button 1 pushes and and the answer on Serial when 2nd button is pushed.
If there is a particular issue that dumbfounds you, then post the code you've tried so far and explain what you want it to do, and what it actually does.
Lastly . Where does the "trinary" come in? A LED is a "binary" device (either On or off). Or are your "LEDs" actually digit display (7-seg) and you are displaying a number in trinary (i.e. using digits 0, 1 and 2 only)?
Edit: The OP edited his question explaining the trenary. So my question here is thus superflous.( I dont like edits to top ost that make my answer/help/clarifications seem idiotic.)