Hello htmanning,
Thanks for using Apple Support Communities.
I understand that you are not getting prompted for your two-factor authentication on other devices than your iMac, as well as having to sign in to your web browser every time. To get started devices that are running iOS 9 or Mac OS X El Capitan are the only ones compatible with two-factor authentication. If your other devices are using those OS versions, then make sure those are under your trusted devices:
View and manage your trusted devices
You can view and manage a list of your trusted devices in the Devices section of your Apple ID account page.
- Go to your Apple ID account page.
- Sign in with your Apple ID.
- Go to the Devices section.
The device list shows the devices that you're currently signed in to with your Apple ID. Select a device to view the model, serial number, and other useful information, including whether or not the device is trusted and can be used to receive Apple ID verification codes.
You can also remove a trusted device. Removing a trusted device will ensure that it can no longer display verification codes and that access to iCloud, and other Apple services on the device, is blocked until you sign in again with two-factor authentication. If you need to find or erase your device before you remove it from your trusted device list, you can use Find My iPhone.
Two-factor authentication for Apple ID
As far as entering the verification code when logging in to web browsers, make sure you select to trust the browser:
When you sign in on the web, you can choose to trust your browser, so you won’t be asked for a verification code the next time you sign in from that computer.
Take care.