And the 28-year-old England international is optimistic he can play his part in bringing silverware back to Anfield, with their last trophy coming in 2006.
Cole completed the formalities of signing his four-year contract yesterday when he passed a medical at the Reds' Switzerland training camp.
Cole admits things have moved fast since he opted to move to Merseyside on Monday instead of remaining in London where he has spent all his life and was attracting interest from Arsenal and Tottenham.
"The last few days have been crazy," said Cole.
"I set myself a deadline to make a decision and when I made it I sent a text to (managing director) Christian Purslow and Steven Gerrard and then turned my phone off.
"It's a dream come true. I've only been here for a couple of days and my head is still in the clouds. It has all happened so quickly.
"I always knew this was a massive club but until you sign you don't realise.
"I have had people on my phone who I didn't even know were Liverpool fans.
"It seems everyone I bump into is a Liverpool fan - it's a huge club."
Cole's free transfer to Liverpool came as something of a surprise as it was believed he did not want to move from the capital.
However, the midfielder is fully prepared for the challenge which lay ahead both on and off the field.
"It's a massive deal. We've just had a young baby girl, but I'm 28 now and I have a good family around me," he added.
"I know people will make me feel welcome and the response I have had has been amazing.
"I just want to get involved. When you move to a new club it's important to immerse yourself into everything about it and that's what I'll be looking to do.
"I know the fans are great with the players and I will fight to try and win trophies for this club."
Cole's signing was seen as pivotal in the club's quest to hold on to prize assets Gerrard and Fernando Torres, who had both expressed their desire to see things improve at Anfield.
Just 24 hours after it was announced Cole had agreed personal terms Gerrard returned to pre-season training after his post-World Cup break and immediately declared his desire to remain at his boyhood club.
And that was just as big a boost to Cole as it was to manager Roy Hodgson, the players and the fans.
"That was the best bit of news I could have had," said the former Chelsea star.
"I know Stevie, he is Liverpool through and through and I know from playing with and against him for the last 10 years what a quality player he is.
"I know how much this club means to him. I think players can play for other clubs but with Liverpool you have to live and breathe it.
"It's that kind of a place. I am excited by the challenge. I am moving my family and everyone is excited and booking their weekends up here. I am going to make it work."
Liverpool also completed the signing of 18-year-old Rangers defender Danny Wilson yesterday.
"This is a massive challenge for me. This is one of the best leagues in the world and I will be playing for one of the biggest teams in the world," said the centre-back.
"It wasn't a difficult decision; there were other clubs interested in me who are in the Champions League but once Liverpool came in for me there was only one team I wanted to go to."
While the pair are moving into Anfield, winger Albert Riera is on the brink of going in the opposite direction with the Reds in talks with Greek side Olympiacos.