The 18-year-old centre-back signed last week from Rangers for an initial fee of £2million in a deal which could rise to £5million.
"Since I have come to the club everyone has been so positive with me," he said.
"It is exciting for me to come to a club this size and now I just want to get on with it.
"I just want to do the best I can. I have not set myself targets as a player.
"If I'm selected for games then great but if not then I'll work a bit harder. Other than that I am just looking to get a good start."
Hodgson felt the club had signed a player who was destined for a great future.
"In Danny I think we have acquired the services of one of the brightest prospects in Scottish football," said the 62-year-old.
Fellow new signing Milan Jovanovic is also looking to take his game to the next level.
"This is a big step in my career," said the Serbia international, who moved from Standard Liege on a free transfer.
"I cannot explain all my feelings but I am looking to the future and I will try to bring something to Liverpool."