How do I track my cat's movements using an AirTag and the Find My app?

Not sure if you are aware of how an AirTag operates.

An AirTag depends on other iPhones in very close proximity to pick up the signal from the AirTag and forward it to the Internet.

If your cat will frequently be within a range of about 20-30 feet from an iPhone, then there is a good chance that you will be able to locate your Cat when you log on to Find My. The more iPhones in close proximity the better.

Now, if your cat slips outside your house, then the chances that another iPhone will pick up the AirTag signal drop dramatically.

For what its worth, Apple does not recommend using an iTag to track pets.

AirTags are decidedly not good at tracking critters.

Cats won’t and don’t get tracked when they’re out of range of an Apple device needed to relay the AirTag position, and that with a range of 10 to 30 meters maybe as much as 100 meters maybe and that under ideal conditions. Maybe.

Cats underneath or inside structures or vehicles will have reduced range.

And with no location available, if the cat is in a location with no Apple devices within Bluetooth range.

Cats are also good at getting out of collars too, and a properly designed cat collar allows a collar-trapped cat to escape the collar. That means an AirTag may be pinging away from within some shed, with the cat elsewhere.

Available location data can be viewed via the Find My app.

Locate an AirTag or other item in Find My on iPhone - Apple Support

In addition to the others, if tracking your cat is critical to you, then you will want to consider getting a dedicated tracker that is based on GPS, not Bluetooth like the AirTags. However, just to be forewarned, GPS trackers tend to be much more expensive, and their batteries do not last as long as those of the tags. It's really a trade-off of cost vs. real-time location accuracy.

A vet tech I know and respect uses a cat tracker from a company called "Moggie."