Can I use the manual option on my Mazda6 to save money on gas?

You can use it to hold each gear longer (better acceleration) and to down shift for braking. Its still an automatic, not standard transmission, the gas mileage losses occur due to slippage of the converter and pumping fluid around throttle bodies, on a standard transmission no fluid is pumped around and there is no slippage loss. No you will not save any money by shifting the gears manually.

there are no throttle bodies in a automatic transmssion, but to answer the original question, let the automatic change when it wants to, use the engineering that went into designing the car.

all the manual option does is let you have more control of your car, but at the same time its really not that different to driving and automatic. my sisters nissan altima has the same feature. nothing special about it there almost a 2 second delay in manual shifting, and when you down shift it won’t shift because of unsafe engine speeds. but really the difference with manual and automatic transmission is about 1 to 2 mpg
accodring to mazda the mazda6 equiped with the 2.3l engine gets 24/32 m/t 24/31a/t; the 3.0l engine 19/27m/t 20/27a/t
as you can see there isn’t much you can of a difference according to epa. but the best advice i can give is to not step on the gas as much it lets me drive an extra day or two without having to fill up.