First-half goals from Marvin Morgan, Conor Hourihane and Rommy Boco put Argyle in control as they bounced back from back-to-back defeats without scoring at the start of the campaign.
Having narrowly avoided the drop for the past two years, Sheridan insists the club can now start to look upwards.
"We know the support we have and we want to do well," he said. "We want to be talking about promotion, not relegation, and if that puts pressure on me then so be it.
"We are a club who can get out of this league at my first attempt in my first season.
"I have to get that into the players and if we turn up and perform like that the fans will support us all the way."
Plymouth had lost to Southend United and Fleetwood in their opening matches, with Sheridan labelling the performances as unacceptable.
But his players responded in the perfect manner to leave Cheltenham still looking for their first win.
"The manner in which we've lost the first two league games wasn't good enough," Sheridan said.
"We didn't deserve anything from either of them, but against Cheltenham, who are a very good home team, we did.
"We got away with one or two things in the first five minutes that we didn't at Fleetwood, but that's football. After that we looked very comfortable and a 3-0 lead is obviously always a good cushion. It was a very good all-round team performance."