City are beginning a new era under Pellegrini following the sacking of Roberto Mancini in May.
Pellegrini has now met most of his squad for the start of a pre-season programme which includes trips to South Africa, Hong Kong, Germany and Finland.
Kompany told the club's website: "I think all of us have done some preparation work during the holidays because the guys look very sharp after a long break, so that's great.
"It was also nice to start work with the new manager and I think his approach with the team today was really good.
"It's important now to try and improve on the last campaign and make sure that whatever the new manager brings in, that we try and make it our own as quick as possible."
City lost their Premier League crown last season and were surprisingly beaten by Wigan in the FA Cup final, events which preceded Mancini's departure.
But since then there has been change with, as well as Pellegrini's arrival, the club signing two high-profile players in Fernandinho and Jesus Navas. Carlos Tevez has also left for Juventus.
Kompany said: "We're at a big club, so the only goal we can have is to win trophies.
"There's about six or seven clubs in England alone in this situation, so it will be very competitive.
"Our objective this year is to win things and to play well together. We all know that we have to be very strong as a team and bond with each other."