iMac A1312 goes straight to macOS Utilities and can’t get it to do anything else

That Mac's two original, grey, System Install DVD and Applications Install DVD must accompany those Macs from cradle to grave. Presumably they became separated in its decade-long journey to Goodwill.

Your next course of action is to purchase a retail, white, Mac OS X Snow Leopard DVD. Apple no longer sells them but you may be able to find one on the used market. Boot from that disc, install that operating system, and update it using Software Update. That will result in OS X 10.6.8, which will incorporate the App Store, which conveys the ability to upgrade to the latest macOS version compatible with that Mac (probably High Sierra). Read How to download macOS - Apple Support. Scroll down to High Sierra 10.13. Follow its instructions (use Safari).

apparently your imac might/might not support internet recovery… here’s the minimum requirements for that; https://www.idownloadblog.com/2016/02/25/how-to-start-up-your-mac-in-internet-recovery-mode/

so, either built in recovery, make a Mac usb bootable stick/thumb drive/flash drive with Mac El Capitan on it… for which you’d probably need the use of/access to/to borrow another Mac, plus a bootable drive maker application/software eg Lion DiskMaker among others… you might be able to make a bootable Mac DVD, assuming your iMac has a built in cd/dvd drive that’s working, if not you can buy one cheaply, then hold down “option” key on startup, put dvd in drive label side up, it should show up on screen, pick dvd, say “yes/ok” and let it do its thing…

Okay, try usual first… reset smc/pmu followed by 3-4 pram zaps on startup, per this link:

https://www.macworld.com/article/224955/how-to-reset-a-macs-nvram-pram-and-smc.html

… you should have wired usb keyboard, use that… restart holding down “D” key for Apple Diagnostics…

if you want to try Internet recovery, make sure Ethernet cable is connected from modem/router to Ethernet port on back of iMac… to try that, restart holding down … see link : Use macOS Recovery on an Intel-based Mac - Apple Support (CA)

you should see spinning globe + progress bar on screen, when bar is full internet recovery will start… if not try recovery partition, hold down Apple/clover leaf/command key + R together on startup to get to that, pick language, continue, to get to Mac OS X tools, move cursor/mouse/trackpad to top centre of screen and click, and there they’ll be… from there you should be able to reinstall the Mac OS..make sure to set your internal hard drive as the boot up/ start up drive then restart after you have reinstalled the Mac OS…

hope this helped you

John B

... tried to reinstall High Sierra but get a pop up that says The recovery server could not be contacted, ...

I just attempted to reinstall High Sierra on an eligible Mac using OS X Internet Recovery. It repeatedly failed with that same message.

After at least several days or weeks of reading reports essentially identical to yours, I am beginning to suspect Apple is no longer interested in maintaining High Sierra over their Recovery servers. That leaves you with the optical installation media options I described in my earlier reply.

Thanks John Galt for the reminder....Yes, he's correct, you do need at least a DVD copy of Snow Leopard (a legitimate one)

to start up/boot up from if nothing else.... those original grey install DVD's might have gotten separated from the iMac, even though they should have been included/accompanied it from beginning to end, they didn't, so are on permanent vacation (M.I.A) so, Snow Leopard DVD it has to be

JB

The good news is that once the App Store is installed (which requires OS X 10.6.8) we can still download High Sierra from the App Store. That had been my next concern, but apparently only the Recovery servers are affected. It may or may not be temporary.

The model number is insufficient information to make that determination. The model identifier is needed: Computers that can be upgraded to use OS X Internet Recovery - Apple Support