Currently, we have only the one computer (Dell GX270) hooked up to the Internet. However, we are bringing in a new computer (iMac) into the mix, and we'd like to keep the Internet connection on both. What is the easiest and cheapest way to do this? (I am willing to switch the one cable from one computer to the next as needed, if possible, as it isn't necessary that someone be on the Internet at the same time on both computers.)

Buy a router.

To be more precise buy a router with an integrated switch which most "home" router already come with and then connect both pcs to the router and the router to the modem. You can also connect the 2nd pc to 1st using a cross over cable if you have 2 NIC's on the 1st pc with the 1st pc being connected to the internet and share the internet that way. So you would need to buy an extra NIC and a few meters of cross over cable.