My '99 GT has 101,000 miles on it. I bought it from a mechanic earlier this fall.

Soon after I started driving it, I noticed that the Service Engine Soon light would come on, and stay on. The next time I would drive the car, it would come on and go back off. This intermittent display has been happening for several weeks.

I replaced the gas cap because it didn't feel like it was sealing. No change. I had the mechanic check the car over, but he said he couldn't find any problems. He did tell me that he had replaced the catalytic converter with a generic aftermarket model because the original converter had rusted out.

I have read some of the past posts on this, but am unsure of what to do next. Any suggestions?

I'd recommend the first thing for you to do is retrieve the codes. By retrieving the codes, you will have a starting point to determine why your computer is not happy with the way your vehicle is running.
If you do not have an Autozone or similar retail parts store that can do it for free, take your car to a dealer or local mechanic. Or buy a scanner for $100 bucks. Wal-Mart sells 'em.
In your case, if I was in your shoes I might get an independent point of view. If you don't buy the scanner, consider paying for the diagnostic so you don't wind up paying for unneeded parts and labor.
In my case, I take my cars to the GM dealer when I can't figure out a problem. I'd prefer to pay the extra $$$ for trained GM techs to diagnose my car instead of a local mechanic guy who "learns" on my vehicle.
Post the codes here in the forum and you'll get some good info/advice from very knowledgable members as to what's happening with your car.

Mark,

Thanks for the advice. I may try AutoZone first, but the dealer is definitely an option.

Before paying the sealer 60 bucks to have your car scanned, go buy an OBD2 scanner for walmart, I think the have dropped in price to 50 bucks. Then you can scan your own codes.

Thanks. I'll check that out.