"I'd be delighted if we can finish fifth or sixth next season and the chairman agrees," he told BBC Scotland.
"But, if you work hard and get a bit of luck, hopefully I can take Morton up to the Premier League in the three years of my contract."
Stirling Albion will join Morton in the First Division after Moore masterminded promotion for the Binos by winning the Second Division.
However, he has opted to leave his old team behind and join the Greenock side that finished eighth this term.
"I can walk away from Stirling Albion with my head held high after winning the title and promotion," Moore added.
Peter Weatherson will be a big player for us but you need to get balls in the box for him
"I'm hoping I can do the same at Morton and maybe get them up to the Premier League after a year or two."
Moore has a big rebuilding job on his hands, with several players moving on, including Ryan McGuffie, Erik Paartalu, Steve Masterton, Dominic Shimmin and Jim McAlister, who has signed for Hamilton in the SPL.
"We've lost quite a few but sometimes that can help you as a manager," Moore explained. "I can go in there and pick my players.
"I felt Morton had too many boys that were one-paced in the middle of the pitch and I like my players to get about the pitch and I like balls to come in from the wide areas.
"I was a winger myself and I like to play with two wingers, but they are hard to come by these days.
"I'm a bit disappointed McAlister has gone but good luck to him because he's a football player."
Peter Weatherson has scored 93 goals in seven seasons at Cappielow and Moore is hoping the big striker can make a full recovery from a troublesome pelvic injury.
"Peter will be a big player for us but you need to get balls in the box for him," he said.
And he is hoping to lure two of his former players with him from Forthbank to help supply that ammunition.
"I'd like to take Andy Graham, the right-back, but he's been offered pre-season by Hamilton.
"I had him for five years and he's the best professional I've ever worked with.
"The other one is left winger David O'Brien. He went to Dundee and it never happened for him but I believe I can get the best out of him."
Source: BBC Sport