Unable to sign into iTunes in order to Update Software

My 2012 iMac has the 10.8.5 software (aka mountain lion) and no matter what, it won’t update. I tried contacting apple, even they couldn’t help. When I try signing into iTunes, it tells me there is no network although there is since chrome works. When I open the App Store, it says it “Cannot connect to App Store.” Apple told me to shut it down and as soon as I hit the power button to hold “command” “Alt” “R” and click the second option, but that didn’t work either. I appreciate any help.