1989 Buick Century hesitation

Starting and cranking are two different things. Are you saying the vehicle cranks good, but takes a few seconds to start? If that is what your saying, the first thing I would check is the fuel system. I'm not saying replace anything, I just trying to understand your problem.

If your saying, when you turn the key to crank, there is no starter action for a few seconds, you might get the battery checked, clean the cables on both ends. If it's an automatic tranny, you might try starting in neutral, see if it makes any difference.