The former Manchester United midfielder signed a new deal with Everton during the week.
On his personal hopes with the Toffees, Garner said: "It's hard when you first come into a club when you've already got so many leaders, so you have to find your feet and find your place – and I feel like I've done that now and it's time to step up the responsibility on and off the pitch.
"I have had to step up to the plate (in the absence of others recently), and I feel like my performances have shown that and shown I can be one of the influential leaders in the team.
"I still feel like there's more to come from me. I'm still only 24 and I've had a lot of experiences already but there's still more that I can do. The obvious one would be goals and assists, which is obviously the one everyone wants to get better at.
"I also want to bring that overall consistency and be a player that people can look at as a leader who is setting the right examples."