Barkley missed the majority of the last campaign due to injury, but the playmaker has been given a chance to impress in Maurizio Sarri's 4-3-3 formation.
The England international has enjoyed a successful pre-season with the Blues, but the former Everton man has acknowledged that he must continue to look for ways to improve under the new head coach.
When asked about his targets for the new campaign, the 24-year-old told The Telegraph: "To keep learning. The new manager obviously has a different style to our previous manager. There will be freedom because you have three men in the midfield.
"You can go forward, get close to the box, have shots, try to create but obviously we have to be disciplined and organised without the ball. I want to be scoring goals, creating goals and, towards the end of the season, there will hopefully be a prize, with a trophy or a couple of trophies.
"One step at a time, but I'm hoping to start the season really strong."
Barkley played the opening 68 minutes as Chelsea registered a 3-0 win over Huddersfield Town at the weekend.