Is Premium and higher octane better than regular gas in Mazda 3, 2012 car?

Octane ratings have nothing to do with fuel economy or power. The octane rating only indicates how much the fuel air mixture can be compressed before it ignites without the spark plug firing in a hot running engine. Some really high compression engines require higher octane fuels so they don't knock when running. If you don't have a sticker under the filler lid that says you have to use one of higher octane fuels, then don't use high-test.
It used to be that using a low octane fuel in an engine designed to use high octane fuels could damage the engine.
Nowadays, engines are all computer controlled and most high octane engines will re-tune themselves on the fly so that they don't knock if the wrong grade of fuel is used.