Chosen Solution

I can backup the 3GS on my Mac tiger but that iTunes does not recognize an iPhone 5 for transfer. My Mac high Sierra recognizes the iPhone 5 and the iPhone 3GS but will not allow a backup of the 3GS.

You’ve got two easy possible ways: - if you have a pc or a mac, you can use iTunes to make a local backup on it, then you plug-in the iphone 5 and store your backup on it. It takes maybe 10/15 mins. If you don’t have a computer you will have to backup all on iCloud and then download it on your newer phone. You can store freely up to 5Gb of data, if you have more you’ll have to pay Apple for a higher storage. Time depends on how fast your net access is.