1998 Chevy Tahoe Fuel Gauge

I hope I'm in the correct category. My question is that every time I start my Tahoe, the fuel gauge needle vibrates very fast, it doesn't move excessively, it stays in one place just vibrates. When I Turn off the engine the fuel gauge needle sometimes goes to the full mark instead of going down to empty as normal. I also noticed sometimes when I turn on the engine the Service Engine Soon light will come on and other times when I turn on the engine it wont come on. (Some times will and some times wont.) Can someone give me some advise as to how I would go about fixing the problems.