Despite decades of attempts to straighten out the connections between printers and computers, it's a situation that always (at least for me) seems to have an extra complication. On Chrome OS, ever since launch it has by default relied on Google Cloud Print and compatible printers to get the job done, but now that version 59 has reached the Stable release channel, it's a little easier to use.
That's because it has a new ability for "Native Printing," which basically means that it can connect directly to compatible printers on your network, without the need for any cloud connection at all. Some manufacturers and third parties have been getting around the cloud print requirement with extensions like this one, but now support is built in and available to everyone.
You'll need to know your printer's IP address to make things work, as well as which protocol it supports. According to Google's FAQ, for most printers that will be IPP, which is what allows many printers to connect to other devices like your phone or tablet. As Chrome OS spreads throughout schools and businesses, the ability to work with existing hardware has become more important, and at least now you're probably not looking at a printer replacement along with your new operating system.