HOUSTON — Growing up in Cuba, Yordan Alvarez was taught that the United States was a bad country. The thought was so ingrained in him that when he was 12 or 13, he said, he skipped English classes at school.

“Why would I go to an English class if I’m never going to the United States?” Alvarez said he told himself then.

Look at him now.

He was the 2019 American League Rookie of the Year, a level he never thought he would reach. He blasts baseballs harder than all but a few other major leaguers, and even sent one over Fenway Park’s Green Monster in Boston in Game 5 of the A.L. Championship Series on Wednesday. He was named the most valuable player of the A.L.C.S. for a Houston Astros team that is World Series-bound after a Game 6 win powered largely by Alvarez’s bat.

He spends his off-seasons in Tampa, Fla., and his two children were born in the country he was told to dislike.

Alvarez, 24, laughs about it now. Maybe he should have taken advantage of that head start with the language of his adopted home.

“When I came to the U.S., that’s when I started to learn,” he said in Spanish, standing on the field before a recent A.L.C.S. game against the Red Sox. “I can tell you that I regret it, but now I can tell you I’m not sure if they were teaching English correctly.”

Alvarez’s story is familiar to many of his fellow Cuban-born players in Major League Baseball, including two Astros teammates, first baseman Yuli Gurriel and the backup infielder Aledmys Díaz. Many escaped the Communist country, often putting their lives in the hands of smugglers or taking harrowing boat rides, or both, to chase their dreams. To play in the big leagues, Alvarez had to leave.

At 16 and 17, he played two seasons with the professional Cuban baseball team in his home province, the Leñadores de Las Tunas. In 74 games in the top Cuban league, he hit .279 and had one measly home run. “And it was an inside-the-park one,” he said.

Back then, Alvarez was known as a nimble outfielder with a good eye at the plate rather than an imposing power hitter. Still, there was potential: Although skinny, the 6-foot-5 Alvarez said he was always the tallest player on his teams. The size, he said, came from his 6-4 father, who also used to play baseball in Cuba.

When Alvarez and his family decided to pursue his baseball opportunities in the United States, he said, he asked for permission to leave Cuba, but was denied. So, in 2015, he went to the Dominican Republic, where he joined his parents and younger brother, all of whom had arrived first.

In the Dominican Republic, where all 30 M.L.B. teams operate baseball academies, Alvarez began working with a private trainer. He said he lifted weights, hit daily from morning to night and overhauled his left-handed swing because it was “never going to hit home runs.” The power slowly began to emerge.

But to sign with an M.L.B. team, Alvarez needed to establish residency in a country, so he went to nearby Haiti. There he ran into Gurriel and his younger brother, Lourdes Jr. — the sons of a Cuban baseball legend — who had just defected from their homeland and were also securing their paperwork in hopes of reaching the major leagues. They kept their happenstance encounter secret.

“I had seen him play in Cuba,” Gurriel, 37, said of Alvarez in Spanish. “He was very young. He was big then, but not as big as he is now.”

After arriving in the United States, Alvarez went to West Palm Beach, Fla., to continue his training and work out for prospective teams. He grew close to an Astros scout, Charlie Gonzalez, who told Alvarez he could picture him in a Houston uniform and who drove him by the Astros’ spring training complex when it was under construction.

Gonzalez was one of the Astros officials who wanted the front office to sign Alvarez, but the organization faced significant penalties for going over their bonus pool limit on international signings. One of the players they had committed to: Gurriel, who had agreed to a five-year, $47.5 million contract.

Instead, the Los Angeles Dodgers signed Alvarez in June 2016 to a $2 million deal. Six weeks later, the Dodgers needed a relief pitcher, so they traded Alvarez, who had not yet played a game in the minors, to the Astros for Josh Fields.

Alvarez shot through the Astros’ farm system. He hit .343 with 23 home runs for the Class AAA team in 2019 despite pain in his left knee that began the previous season and flared up as time went on.

“My goal was reaching the big leagues,” he said. “But I also thought to myself that I would never reach it if I was messed up. I needed to keep playing.”

Despite the balky knee, Alvarez accomplished his dream on June 9, 2019, at 21, after the Astros’ illicit sign-stealing had ended in M.L.B.’s eyes. Adrenaline, he said, masked the pain, and he continued pushing himself. He hit .313 with 27 home runs in 87 games as the Astros’ primary designated hitter, and he helped them reach the World Series, where they fell one win short of a title to the Washington Nationals.

Playing on a compromised leg for so long, Alvarez said, led to overcompensating with his right knee, and that caused damage there. Finally, after playing in two games in the pandemic-shortened 2020 season, he could not take it anymore and had surgery on both knees (a patella tendon repair for one and a cleanup of the other). He missed the rest of the year.

With stronger, healthier legs this season, he has felt a difference. Only seven major league players consistently hit the ball harder than Alvarez, including Aaron Judge, Giancarlo Stanton, Vladimir Guerrero Jr., Fernando Tatis Jr. and Shohei Ohtani.

A common thread among them? They are large human beings. Besides being 6-5, Alvarez is 227 pounds.

“A lot of big guys can’t hit,” Manager Dusty Baker said recently, before referring to Alvarez. “He has good vision. He has good balance, especially now with his legs that they’re good. Balance is the key, and he can run. Big joker can run, and he thinks he can hit.”

In 144 games this season, Alvarez hit .277 and led the Astros with 33 home runs and 104 runs batted in. He strikes out a fair amount, but when he connects, he hits it hard, far and in the air. Case in point: Against Red Sox starter Chris Sale on Wednesday, Alvarez’s Green Monster shot came when he flicked a 94-mile-per-hour outside fastball to the opposite field, hitting the seats above the famous wall.

Astros shortstop Carlos Correa called Alvarez “a natural hitter.” Gurriel said Alvarez had a maturity at the plate that belied his age. “That makes him very special,” he said.

Inside the clubhouse, teammates said Alvarez also defied expectations. Perhaps because of his imposing stature, his facial expressions or the language barrier, people often think he is a very serious person, Alvarez said. His wife, Monica, sometimes tells him to smile, he said; otherwise, he appears to be in a bad mood.

“I do like to joke around,” Alvarez said. Correa added: “That guy doesn’t shut up in the clubhouse. He looks quiet, but don’t let him fool you.”

It helps, though, that Correa, a Puerto Rican, is bilingual. So is Alvarez’s wife, who was born in Cuba but came to the United States at 5. Alvarez said she has helped him a lot with his English. If it is baseball-related, he said he understands most of a conversation, but one of his goals is to do a better job of learning the language away from the field.

Another of Alvarez’s dreams — one he never imagined would be possible as a child growing up in Cuba and skipping English classes — is also in the works. He said he was in the process of securing the paperwork to get his parents, who are in the Dominican Republic, to the United States so they can watch him play in person for the first time here.

“My mom would, for sure, love it,” he said. “But my dad, who played baseball, would love it the most.”