1993 Pontiac Grand Prix engine not staying started initiall

When starting engine it won't stay running right away. Have to keep excelerator down for about a minute before it will continue to run on its own. Also seems to emit some excess smoke initially. Runs smooth when cruzing but sometimes seems starved for air or fuel when excelerating as it will cough but runs fine when reaching cruzing speed again. My cousin seems to think it is either the fuel filter, or fuel pump that is causing the problem. I would be interested in what your opinion is.