I'm going to try to give an answer that applies both to the NYT app and to apps in general. The short answer is make sure that you setup both phones with the same Google account and sign into your NYT account in the app.
If an app is properly designed (and I would assume that the NYT app is) then user settings, such as favorites, should get transferred to your new phone automatically.
Some apps use the data backup/transfer functionality that is built into Android, which relies on you setting up both phones with the same Google account.
Some apps instead sync user data to their own servers, which typically relies on your signing into the app with their account. The NYT does have its own system of users accounts and allows you to sign into your account in their Android phone app (oddly, this is missing in their Android tablet app).
If an app does not implement either of these then you should give it a low rating (using the built-in Android data backup/transfer functionality is super easy so apps really should implement it).
The principle exception to this is cases where your data is very large, i.e. media files, in which case the app should provide functionality or instructions to assist you in moving your data.