Grog is typically beer and usually not a strong one. It originates from what sailors called watered down Rum.
A day on the Grog is one of those drinking sessions that is leisurely in pace. One of those sessions that quietly does it's job.
A day on the Grog is different to a day on the Piss (as that's going at it hard and fast), it's a day where you and your friend(s) settle into a nice easy cadence of drinking, and by late afternoon you find yourselves rather content, a little bit sloppy and fairly drunk.
"Bloody hell boys, it's only 5 in the afternoon and you're swaying all over the place. Good day on the Grog or what?"
"Alright Dai? Shall we go watch Wales play Rugby and have a day on the Grog"
"I cant have a mad one today boys, last night was heavy. Lets just have a nice quiet day on the Grog"