Wallace was introduced to the media on Wednesday after signing a five-year contract. The deal is worth $60 million and includes $30 million in guarantees, sources told ESPN NFL Insider Adam Schefter.
The former Pittsburgh Steelers receiver was asked if he could give a time frame for the Dolphins to compete for the AFC East crown with the Patriots and vie for a Super Bowl title.
"Hopefully we can compete this year. That's why we're in this. We're in this to compete every year," he said. "We expect to compete as soon as we step on the field. From Day 1, we're going to compete with everybody and we're going to let the chips fall where they will."
Miami on Wednesday also announced the additions of linebackers Dannell Ellerbe and Philip Wheeler. Ellerbe signed a five-year deal worth $35 million, according to the Baltimore Sun. Sources told ESPN that Wheeler received $26 million over five years with $13 million guaranteed.
Dansby, a nine-year veteran, and Burnett, an eight-year veteran, started side by side every game the past two years for Miami.
Wallace said it was hard to leave Pittsburgh but he had to look out for himself when he hit free agency.
"That's all I know so it was definitely hard for me [to leave the Steelers]. But as we all know this is a business and you have to make the best decision for yourself. I'm excited to move forward," he said.
He said the biggest adjustment would be adapting to playing football in the South again after playing four seasons in Pittsburgh. The Steelers drafted Wallace in the third round (84th overall) in 2009 out of Mississippi.
He joins a Dolphins receiving corps that had just three combined touchdown receptions last season. Wallace had eight touchdowns last season for Pittsburgh.
"Everything is in the past with those guys and we're all a part of the same group now," he said. "I'm going to do my best to motivate those guys and I think we'll see more touchdowns from everybody."
He said he has already spoken with his new quarterback, Ryan Tannehill, and had dinner with him Tuesday night.
"He picked the check up, so he's a great guy," Wallace said with a chuckle.
Information from ESPN NFL Insider Adam Schefter was used in this report.