Mac OS Yosemite

With a late 2013 iMac you can upgrade as far as Catalina. Depending upon the RAM and number of old applications you may wish to try something intermediate instead, particularly since that age iMac likely has a slow rotational disk drive which is not easy to change for a faster SSD. Catalina will force your Mac to use APFS drive formatting which can be harder on rotational drives. My inclination would be to use High Sierra or Mojave first and see how that works out. They aren't the newest but unless you are trying to download something that insists upon them they would probably hit a sweet spot.

Click on this link to read how to get different versions of macOS - https://support.apple.com/HT211683

- Make sure you have more RAM (memory) than what is noted as "minimum" for any particular system or your computer may run slowly.

- You may find you need to use Safari to get links for High Sierra and later using the Mac App Store application to open a functioning page.

- Sierra, El Capitan, and Yosemite download as a .pkg. Click on this to have a converted version placed in your Applications folder as an "Install OS..." application. Run the installer and follow the directions.

- Upgrades may take some time (up to an hour or more) to install, with restarts and blank screens.

Before upgrading, back up your data in case something goes wrong. See this support document: Back up your Mac with Time Machine - https://support.apple.com/HT201250 or use a cloning tool.

Check if your old apps will still work with with the OS you want to install - https://roaringapps.com/apps This is particularly critical if you are upgrading to Catalina as older 32 bit apps will no longer work. To check for those, try this 32 bit application scanner application --> https://www.stclairsoft.com/Go64/