Here is a State Transition Diagram which I think implements your requirements. There are a total of six inputs, three Buttons B1,B2,B3, and three limit switches mounted at the parked position for each floor L1, L2, and L3.
There are seven states, where the elevator is stopped at FIRST, SECOND, and THIRD; is moving up during states UP2 and UP3, and is moving down during states DN2 and DN1.
It takes three flip-flops to represent seven states (with one possible state unused). I have assigned state values so as to make them easy to use to control the motor. Note the bit pattern 100 and 101 occurs when the elevator is moving up, while 011 and 010 occurs while the elevator is moving down. 000, 001, 111 occur while the elevator is stopped. If you look at the leftmost two of these bit patterns, they can be used to control the motor directly, where 10 means up, 01 means down, 00 or 11 means stop.
I'm out of time tonight. You have some reading to do. Read up on "State Transition Table or Diagram", State variables, Present state/Next State, etc.