The players were made available at a news conference as the Astros opened spring training in West Palm Beach, Florida.
"I am really sorry about the choices that were made by my team, by the organization and by me. I have learned from this and I hope to regain the trust of baseball fans," Bregman said.
Altuve said the Astros had a "great team meeting" on Wednesday night and said the "whole Astros organization feels bad for what happened in 2017."
"[I] especially feel remorse for the impact on our fans and the game of baseball," he said.
The apologies are a change from their responses at the team's fan fest last month, when neither player showed remorse for his actions.
More players were made available later in the team's clubhouse. Yuli Gurriel said "no one put a gun to our head" and "it would be a lie to say that one or two people are responsible."
He and George Springer both said "we are all responsible." Springer said Carlos Beltran and former Astros bench coach Alex Cora, the two people singled out in MLB's investigation as the ringleaders of the scheme in 2017, "were great to us."
Justin Verlander, who joined the Astros in August 2017, said he wishes he had done more to try to stop what was happening.
"Once I spent some time and understood what was happening, I wish I had said more. I can't go back and reverse my decision. I wish I had said more, and I didn't," he said.
New Astros manager Dusty Baker said he hopes his players will be forgiven by fans and other players.
"I ask the baseball world to forgive them for the mistakes that they made," he said.
Astros owner Jim Crane, as he did last month after Major League Baseball released its findings and punished the organization for the sign-stealing scheme, apologized and vowed "that this will never again happen on my watch."
He pointed out that he went beyond MLB's decision to suspend manager AJ Hinch and general manager Jeff Luhnow by firing both men, and Crane said he agreed the players should not be punished. He called them "a great group of guys" who didn't get the proper guidance from Hinch and Luhnow.
When asked whether the Astros should keep their World Series championship from 2017 after beating the Los Angeles Dodgers in seven games, he pointed out that MLB made it clear Houston was keeping the title and said he agreed with that decision.
"Our opinion is that this didn't impact the game. We had a good team. We won the World Series and we'll leave it at that," he said.
He said the Astros "cannot change the actions of the past but [are] fully committed in moving forward in the right way."
Asked about Crane's comments, Dodgers manager Dave Roberts made it clear his team isn't dwelling on the past and that its focus is on the 2020 season, especially after trading for Mookie Betts and David Price.
"... Our guys are really, really focused on '20. Obviously there have been some good things that have happened for us in the last couple days," he said.
Carlos Correa said the sign-stealing system was not as effective in the 2017 postseason because opposing catchers were using multiple signs to tell their pitchers what to throw.
"We feel bad and we don't want to be remembered as the team that cheated to get a championship," Correa said.
"What we did in '17 was wrong. ... ," Correa added. "I'm going to be honest with you: When we first started doing it, it almost felt like it was an advantage. ... But it was definitely wrong. It was definitely wrong and we should have stopped it at the time."
Crane was asked by reporters about speculation that the Astros wore buzzers under their uniforms in 2019 to steal signs. MLB has said there is no evidence that buzzers were used by the team.
"I truly believe there were no buzzers, and I don't even know where that came from," he said.
Asked whether the Astros cheated when they used video to steal signs in 2017, Crane replied: "We broke the rules. You can phrase that any way you want."
Commissioner Rob Manfred, speaking in Kansas City, said of the Astros' apologies: "Look, I think it's important for the front office, the players to take responsibility for what happened, and to express remorse to the fans, the other teams and people who are really invested in our great game."
ESPN's Alden Gonzalez and The Associated Press contributed to this report.