Q: iPhone 6 won't connect to iMac properly

most likely issue can be, or that your phone connector, where the cable goes into has a bad contact, but most of the time it is debris in the device, i had this issue aswell bought a new toothbrush and used it to clean the port, now it is working like a charm, most of the time it is debris in the lightning port of your iphone. if that still does not resolve your issue you should contact apple care, also try it on a different computer if it has the same issue.

but my guess it is debris

do you have the same issue with other devices (MBP or other iMac)?

Please check the SMC and EFI version of your iMac according to bellow link.

EFI and SMC firmware updates for Intel-based Macs - Apple Support

since your iMac is new I assume you are not connecting the cable properly or usb port is too tight. Please nudged the usb cable in the port little and try.

There will be only option remains in this case. The doc connector issue but before confirming that, please test the device with an other mac also.