"We did well last season and the season before," he said. "Now we need to push on again and I think that will happen.
"If we can get a good run going and make our home form a bit better than it has been, I'm sure we can do it and have another good season."
Stoke finished 12th in their debut season in the Premier League after winning promotion from the Championship in 2008.
Tony Pulis' side then became only the second team in Premier League history after Blackburn in 1993/94 to improve both their position and points tally at the end of their second top-flight campaign.
Stoke have yet to make any signings during the summer, although Shawcross thinks the club are now looking to attract a better standard of player.
"We've established ourselves as a Premier League team and we can move on to bigger and better players," added the former Manchester United trainee.
"At the moment there's a lot of paper talk and the manager usually leaves it quite late, so don't be surprised if you don't see any new players until just before the start of the season."
Source: BBC Sport
Source: BBC Sport