The Potters are unbeaten in nine matches and lie eighth in the Premier League table, just five points adrift of fourth place, after a 3-1 defeat of Liverpool on Boxing Day.
Walters bagged a brace in the victory but the Republic of Ireland international warned: "We can't get too ahead of ourselves.
"We've got Southampton and then (Manchester) City on New Year's Day, and we have to keep going and keep our feet on the ground. As long as we do that, we will be all right."
Walters was a boyhood Everton fan, making his goals against Liverpool all the more enjoyable, but he was also keen to pay tribute to Kenwyne Jones, who scored himself and set up his strike partners' second.
"All my family will have been cheering," said Walters. "It is nice to score against Liverpool being a blue.
"But whoever scores, it doesn't matter, it is a team effort. As long as we spread the goals throughout the team and we keep winning we will be happy.
"Kenwyne has done excellently. He is such a threat in the air and holding it up. When he's on it, he is really on it.
"Kenwyne or (Peter) Crouch, whoever is playing, are such a handful up front, and it is nice to have two good players like that."
Stoke boss Tony Pulis agrees with Walters that the side need to remain focused.
"The first thing I said after the (Liverpool) game was that it was a fantastic performance, and you couldn't take that away from us," said Pulis. "But our toughest game this week will be Southampton, because of the expectations of everyone around the place.
"Forty points (is the target) as quick as we can and then we will look at it. That's the way it has always been at this club."