How do I sign into Apple TV for my free month? I’ve been using every password I know. Can it be set up with my Apple password.

Users are supposed to sign in to the Apple TV app with their Apple ID account. That is how the app knows if a subscription applies. Your Apple ID account would have a password of your choice. If you forgot your Apple ID password - Apple Support.

If you get an error message when trying to sign in, then what does it say exactly?

To get a free month, you would have to find a company that makes an offer for a free month of Apple TV+ (note the Plus, to indicate the streaming service of Apple Originals).

• Apple offers a free week trial to new subscribers, or an extended 3 months trial if you buy an eligible Apple device. Redeem that offer on the new eligible Apple device.
• Partnering companies may have other offers for Apple TV+, with other durations, and other conditions, and other redeem procedures. Often through their site, their app, or their device.

Unfortunately, users cannot do (financial) transactions in the Apple TV app on Fire TV at this time [2]. That means that any subscribing (or buying or renting) will have to be done on a different supported device. Once completed, the paid-for service, or free trial, will be available on any or your devices with the Apple TV app, when signed in with the matching Apple ID account.