The 31-year-old goalkeeper believes the Blues must win another major trophy under Jose Mourinho but is aware of the growing competition they will face for the Premier League title
The Blues became the first-ever Champions League holders to be eliminated from the group stages last season and had to settle for a place in the Europa League, which they eventually won.
Chelsea have also struggled to challenge for the league title since 2010 and Cech feels that needs to change next season, even if it means competing with five other teams.
"[Manchester] City keep spending money and bringing in players so obviously they have the intention of challenging for the title," he said.
"Manchester United will challenge for the title and I believe that Liverpool and Tottenham will want to improve.
"Arsenal will want to prove everybody wrong. They will be wanting to prove that they are still hungry for success because the manager is under pressure for results.
"We need progression after last season for it to be a success. Progression means doing better in the Premier League and doing better in the Champions League. We need to win something."
The Czech Republic international went on to explain that he will also be under pressure to deliver, especially since the arrival of Mark Schwarzer from Fulham earlier this month.
"I am under pressure so I obviously need to be at my best to keep my place – if I don’t do that, I am not going to play in the Premier League," Cech added.
"If you have fear you might as well stay on holiday and not come back – it is a waste of time and energy.
Jose Mourinho's side will be Manchester United’s first visitors of the new season on August 26 and Cech is hoping Chelsea put previous Old Trafford results behind them and come away with three points.
"We surely want to improve on the last two seasons," he continued. "When United didn’t play particularly well, they always managed to get a result. I think the start of the season will be important.”