The Dons avoided the drop on a dramatic final day thanks to a 2-1 win over Fleetwood which sent Barnet down instead.
But Moore feels that with a few new faces over the summer the club can push on and challenge at the right end of the table.
"Wimbledon is a great club with great expectations and I think we should be looking to climb the league now," he told the club's website.
"Everyone can go away and enjoy their summer, but we need to come back in pre-season and make sure everyone works hard.
"We have to try and hit the ground running because the key in this league is to get a good start. You can then gain momentum and keep winning.
"It is going to be tough to try and reach the play-offs, but if the gaffer recruits right in the summer then I cannot see any reason why we cannot go for it.
"Neal (Ardley) came in and brought his own players to the club, but the team spirit has been great. One of the first things the manager talked about when he came in was the team spirit and you can see we have that at the training ground every day.
"Having been at the club three years, I am delighted that the club stayed in the Football League, but we need to push on now. Hopefully, we are not in the same position this time next year and we can go for the play-offs."