Is there an Oreo custom ROM for the S3? If so you can install that in the same way that you installed Nougat (since I'm guessing that was also a custom ROM): back up your apps and data, go into your custom recovery, back up your old ROM (in case the new one doesn't work), factory reset (to clear data that will not be compatible with the new ROM), install new ROM, reboot, restore your apps and data.
Technically rooting is modifying the ROM to allow you to grant admin privileges to apps, and the custom recovery is separate (though usually installed as part of the process). Once you have a custom recovery you can use that to install new ROMs without having to do anything else (and custom ROMs are almost always rooted already).
Of course the first question is most important: has anyone built an Oreo ROM for the S3? There's no generic OS that can be installed, it has to be built for the device, so it's only possible if someone has done that.