The former Atletico Madrid striker made a £32 million switch to the Premier League, after he was a big part of the La Liga champions' success last season - as they broke the duopoly in Spain, and were seconds away from UEFA Champions League glory.
Costa, 25, said he is keen to repeat his league success in England, after joining Jose Mourinho's side.
"We have an outstanding squad and if we can all stick together, pull in the same direction and remain focused I don't see any reason why we shouldn't be going to the final stages and fighting for every trophy available," Costa told Chelsea's official website.
Costa said his place in Chelsea's starting XI was not assured, believing he has to fight fellow Spain international Fernando Torres for the number one position leading the line at Stamford Bridge.
"Firstly, it's important for me to respect the situation I'm in," Costa said.
"I've come here to have a great fight professionally with Fernando Torres in order to get into the team.
"Working with players such as those here will definitely aid my game and I know I'll get chances.
"I look forward to playing with them but the most important thing is getting into the team and that's not a foregone conclusion."
Costa added Torres and Cesar Azpilicueta were beacons of information prior to sealing his move to London.
"Of course, I spoke to them (Torres and Azpilicueta) during the World Cup at various times," he said.
"When I knew I was making the step to come here I took advice and asked what to expect.
"They told me it was an outstanding club with great staff and a great structure to it.
"They told me it would be a place where I could enjoy my football and improve my own game."
