I have a game I wanted to play on a different device, my new iPhone, but it isn't that new and I already have too much stuff on it to even reset it.
How do I move that app? And remember, no resetting.
You can download the app again from the App Store under the Updates tab > Purchased and searching for its name.
Your data/progress in the app will not transfer unless:
Otherwise, your progress in the game will be lost. There is no way to transfer data for a single app between devices.