How do you fix a broken gas gauge on a 1989 Pontiac Bonneville?

The question you ask is too involved for a proper answer. You need to go to a car parts store, and get a repair manual for your car. They cost about $16.00 Or, go to a Public Library

U disconnect the wire @ the sending unit-the gauge should either go 2 empty or full THEN ground the sending unit wire & the gauge should do the opposite IF the guage does this then U need A SENDING UNIT-NOT A Guage.check with the key on 4 a short period of time.BE VERY CAREFUL IF ITS THE SENDING UNIT