Thanks folks! I'll definitely do the acid flush with the old radiator in place, and flush well afterwards.
The car has actually had no problem keeping itself cool. I'm just doing it because it looks like it needs it, and I heard it's important to do periodically.

I ordered a bottle of Shout Advanced Gel for the degreasing, and two pounds of citric acid for the flush.

My plan was to drain the radiator and block, then remove the thermostat, remove the top and bottom hoses on the radiator as well as the heater hose, then flush thoroughly with water by running hose water through the heater hose and through the top hose on the radiator.

Then reattach the hoses (leaving the thermostat out), mix the Shout with 2 gallons of hot tap water and dump that in the overflow tank. Run the car for 30 minutes with heater on, then let cool down. Drain, refill with hose water, run engine a few minutes. Do the drain-refill-run two or three times or until the water is not soapy.

Then reattach the hoses (still leaving the thermostat out), then mix the citric acid with 2 gallons of hot distilled water and dump that in the overflow tank. Run the car 30 minutes with heater on, then let cool down. Drain, refill with hose water, run engine a few minutes. Do the drain-refill-run two or three times or until the water is not slippery feeling.

Then install the new radiator, reattach the hoses (and the thermostat), fill with Zerex G-05.

I'd welcome any input into this plan! I'm a beginner, so don't hesitate to point out something that would be obvious to an experienced wrench wrangler.