The 31-year old joined Valencia from Wolfsburg in 2010, but fell out of favour under Unai Emery last season, despite making 31 appearances in his first campaign with the club.
When asked about where his side hope to finish in La Liga this season, Costa said: "The aim as a team is to make the best possible position in the table. Last year we finished in third place.
"We already know it is so hard to be at the same level as Real Madrid and Barcelona, but we are expected to be a successful team and the footballers and the supporters of Valencia want to be third, for sure, and hopefully get a trophy for the club."
The defender will be hoping to make a good impression with new boss Mauricio Pellegrino, who replaced Emery back in May.
When asked about whether there were any similarities between the two, Costa gave a detailed response.
"It's very early to talk about the differences between the two coaches," he said.
"I spent four years with the previous coach and I've only been with Mauricio Pellegrino since Monday.
"Pellegrino is a coach who likes to play good football, hold the ball, pressure your opponents when you lose the ball, and make a strong team. That's the main thing, and Pellegrino has said this.
"Unai Emery was similar but with other characteristics. It's too early to speak about the new coach."
Despite losing Jordi Alba to Barcelona, Los Che are continuing to strengthen their squad during the transfer window, and have already completed the signing of Fernando Gago from Madrid.