Johnthestrugglingdirector wrote: … how should I save them? This old iMac is looking a bit sketchy and I'd like to get these old images off before it's too late.
Right! You don't say-- is the Sierra Mac using the Photos app, or is it using iPhoto? How big is the Library?
In either case the very best thing to do is to copy the Library to an external drive. The Photos Library is cleverly named "Photos Library.photoslibrary," and it's in the Pictures folder. (The iPhoto Library is similar.) Just drag it to an external drive. The best format for the drive is Apple's APFS, and it must be formatted in either APFS format or Mac OS Extended (Journaled) format to be safely run the Library. Storing the Library in other formats is OK, but running a Library from a non Apple format can corrupt it.
A newer Photos app should be able to run an older Library, but it will have to update it first. You should do this on another copy of the Library, just in case. Once updated, a Photos Library can't be run from an older version of Photos.
Macs can have multiple Libraries, and you can just double click on one to use switch to using it. You can combine Libraries-- you can use File>Import to bring in pictures from another library. However, you don't get albums and folders that way.
For dealing with multiple Libraries, importing albums as well as pictures, and even merging two libraries into a combined one, we use the trusted 3rd party app PowerPhotos ($35) which can help with lots of other things as well.