Their 3-1 win at home to Swansea has set up a tough quarter-final at Arsenal - who denied the Toffees another derby at Anfield by beating Liverpool 2-1.
The Toffees last trophy was their FA Cup win in 1995 and with the backing force of manager Roberto Martinez, who won the competition with Wigan last May, Coleman is determined to bring back the Cup to Merseyside.
"We want to go as far as we can in the cup," said Coleman.
"It is going to be a difficult draw but you have to beat them all at some stage if you want to win and there are some good teams still in it.
"I've been here five years now and I've not won anything so it would be nice to win and there are other lads who have been here longer who want to win it as well.
"It would lift the club as a whole. Winning an FA Cup is massive for any club and it would be as big for us and a big stepping stone for other things."
Everton will hope by the time their tie comes around Arsenal will be distracted by a Champions League second-leg clash with Bayern Munich a few days later.
And even though their quest for the FA Cup could negatively impact on the Toffees' attempts to keep in touch with the top four, Coleman believes their squad has enough depth now injuries are starting to ease.
"We have Rom (Romelu Lukaku) coming back from injury so we are starting to get stronger after a few injuries with Antolin Alcaraz nearly back as well," said Coleman.
"We have got a good squad and for the things we are looking for this season we need a big squad."
Coleman also believes that the addition of on-loan Monaco striker Lacina Traore will give Everton a welcome boost up front.
"Lacina has taken a while to get in because his injury but he has been brilliant and is a great asset to have on the pitch - and I am sure that goal will do his confidence the world of good," Coleman said.
"He did well and thankfully he got a goal to get him going."
Martinez is also pleased to have a squad that is virtually back to full fitness.
"We are strong in numbers now," Martinez said.
"All of a sudden we had a couple of players back and Antolin Alcaraz could have been available to be in the squad so I am really excited.
"It is about having everyone ready and I am really pleased with the numbers we have and I feel anyone is capable of coming on and making a big contribution.
"That will be the key from now until the end of the season."