I have a Hampton Bay ceiling fan with three wires emerging from the top: white, black, red, and green. Presently the white, black, and green are hooked up to a switch and the fan works fine. The red is capped. The fan also has a chain which I intend on keeping on the medium setting indefinitely.

Emerging from the fan's light housing are two wires, black and white. If I follow the black wire into the housing, I can see a red wire as its source. I tested the black and white wires from the light housing with a multimeter and using ACV and there is no current. Based on all this, I presume the red wire from the top of the fan that is capped is what powers the light. There is a chain for the light which I intend on keeping "on" indefinitely.

I'd like a single wall switch to turn on the fan and the light simultaneously. If from the top of the ceiling fan, I connect the black and red wire together to the black from the ceiling, will I accomplish what I set out to do in a safe manner. I'm a little reluctant to just test it for fear that I'll fry something.

You can tie your red and black from fan to ceiling black, both will be switched together.
You can turn either off by pull chain.

You both are correct.

Ceiling fan wiring - Ceiling Fans N More