I know nothing about fountain pens; I found two pens in storage, a Waterman and a Parker, which have been left dry for years. I bought new cartridges at Staples and stuck them on the nibs, and . . . nothing. What do I need to do to clean them and/or get ink flowing again? Thank you for relieving my ignorance, I'm sure I'm in the right place.

You probably need to soak the nib sections in cool (never hot) water for a while to dissolve any ink that's dried up inside. If you have a rubber-bulb ear syringe, you can force water through to speed the process.

Welcome to FPN samspade

Richard offers good advice.
If just water is not giving good results then try flushing through with soapy water (dish washing detergent) to a weak soalution.
Also if you are using cartridges try to get the converters instead. When filling with ink the bi-directional action of the ink flow tends to keep the ink feed clearer than with cartridges. Second with a converter you can flush through the pen with water much easier. I have found that you tend to get huge chunks of dried ink being flushed out that soaking alone does not budge.

Good luck and best of enjoyment with using your pens!

Armchop