i have provided the gear on/off patterns
Where?
Have you written any code to test that the relays operate? Have you written any code to prove that setting three relays to known configuration makes the solenoids actually shift the transmission?
Writing code to read the switches, keep track of what gear you are in, preventing going below 0 or above 3, and setting the relays correctly (one it is known whether HIGH or LOW is appropriate for each relay for each gear) is pretty easy.
Showing data on the LCD should also be pretty simple, IF you just want text. If you want graphics, and you want the Arduino to also laugh like a maniac when you miss a shift, that could be a bit harder.
Did you mean LCD screen? 128 by 64 LEDs?