I have a 97 Cutlass Supreme and I want to do a radiator flush.

I'm not really much of a car person, but I'm a pretty quick learner.

What's the best way to do it (besides bringing it in)?

Welcome to AF !

The best way is to first remove the thermostat so you dont hav to wait till it warms up to start circulating.

Then just throw in a bottle of radiator flush(like prestone or any other name brand) let it do its thing. Doesnt hurt to go over on the time its run/cleaning.
Usually it says 20min, wont hurt to do 30min, especially if you hav alot of grimey rusty stuff in you system.
If its real nasty looking use 2 bottles of cleaner.

Then you just drain and fill 3-5 time to rise out all the cleaner and biuld up. I beleive you radiator drains on the driversside.

Heres a link that has some additional info with some pics, that always helps :D It says 96' but its all realavent to what you need to do, and it has other helpfull info.

cooling system (http://www.autozone.com/servlet/UiBroker?ForwardPage=/az/cds/en_us/0900823d/80/0e/c4/d4/0900823d800ec4d4.jsp)

Hope that helps.