Check Transmission indicator

What you need is a scan tool that will read transmission codes.Not all code readers do transmission codes.You more than likely have a sensor out of range.It may be worth a trip to the dealer,they should be able to read the code for free.Not like it takes more than 2 minutes to do.

Thanks. Later in the day, the car now "kicks" as it shifts gears (it's an automatic transmission). Not too badly, but it's bothersome. I checked the fluid again, but it's all the same. I can't imagine what brought this about. The car was working fine for 20 minutes this morning. The problem was very sudden.

Without knowing what the trans codes are,you would need a crystal ball.

Like Boghog says...the codes are very necessary for a correct diagnosis.
When the "Check Transmission" light comes on it means to have it serviced ASAP by a qualified technician with a capable scan tool.
It has detected problems that can and will destruoy your trans in short order. It could be over heating or slipping or just a sensor/connector that has gone bad. The "kicks" shift you feel is the trans trying to protect itself from further damage by increasing the internal pressures so the clutches don't slip.

Couldn't have said it better Jim.

I took it to a transmission shop. The guy said it was overheating, but he wouldn't give me the codes. Still trying to find the problem.

I would not use that shop for the repair if they are gonna be that secretive how could you trust them with a repair that could be from a 2.00 fuse to a 3k trans rebuild.With the trans light on line pressure will be high to prevent any further damage internally.I would try another shop or disconnect the battery to clear codes,drive it and see what happens.If it is a hard fault it should turn the light on be the second key cycle.