“I wouldn’t complain if I got stationed here,” Mr. Warren said. “I’d have fun. But as far as retiring here? I don’t see myself doing that.”  As they moved deeper into the square, Mr. Warren said his childhood involved a lot of time playing video games like “Spyro the Dragon” and wandering in the woods. “I would disappear for hours,” he said. Although he loved the outdoors, he was never much of a hunter. “I’d always fall asleep in the deer stand,” he said.  A Statue of Liberty figure on stilts emerged from the crowd.  “The first time I came here, I thought it was mind-blowing,” Mr. Warren said. “I won’t lie: I fell in love with it. But the more I grew up, the more I realized it’s nice being able to lay out and look at the stars. You can’t really do that in New York. It’s too bright.”  A few women wearing American-flag body paint and not much else frolicked nearby.  “I feel really patriotic right now,” Mr. Warren said.  “Is this normal?” Mr. Vest asked. “Is this how it always is? Sensory overload.”  In the quiet of Two Boots Hell’s Kitchen, a pizzeria on Ninth Avenue, Mr. Vest ordered a slice with pepperoni and a root beer. “I’m fairly new to the Navy,” he said. “I’ve been in about nine months. I started off with the nuclear program, but I couldn’t quite cut it there. It was hard. But I’m happy with where I am now, working to be a radio man on submarines.”  Mr. Warren took a swig of Sierra Mist. “I’m a navigational,” he said. “Basically, I’ll tell the boat where to go, when I finish school.”  The three stripes on Mr. Vest’s uniform signified his rank: Enlisted-3. “Because I don’t have any stripes, that technically means he outranks me,” said Mr. Warren, who is classified as E-1.  At boot camp in Great Lakes, Ill., they learned how the Navy works: ranks, customs, comportment, how to moor ships and fight fires onboard. They did a lot of push-ups, too. “We had a competition to see who could do the most,” Mr. Warren said. “I did 120.”  Mr. Vest remembered getting in trouble one time for not writing precisely enough in a log book: “Our chief started yelling at me because I wrote my M’s wrong. You’re supposed to bring your M’s all the way up, all the way down, all the way up and down again. I had to do push-ups for bringing the middle part of my M halfway down, instead of all the way down.”  So far the two seamen have spent about an hour each inside submarines. But they have spent many hours on the outside, scrubbing, chipping off rust and adding coats of paint.  If all goes to plan, they will learn where they are going to be stationed in the fall and leave the school at Groton within a year. Mr. Warren said his first choices are Guam and Hawaii. Mr. Vest has his eye on Bangor, Wash. “There are good outdoor activities, like hiking,” he said. “A lot of coffeehouses, and I’m a big fan of coffee.”  Asked what they would be doing now if they were back home, Mr. Warren said: “I’d be at the lake right now. Party at the lake. I haven’t been home since boot camp.”  Mr. Vest said: “When we’re on leave, I always go home and see my family. I miss them.”  After the pizza slices arrived, Mr. Warren said there was not much to do during downtime in Groton. “If you go to the mall, the girls are, like, 17 and under. So stay away from the mall. If you go to the bars, you have to be very careful. Groton is a military town, so if you find someone to have fun with, there’s a very big likelihood that she is related to someone in your chain of command, which ends very badly for you.”  During the few hours he had been in New York, Mr. Warren had received calls and texts from a woman he had dated in Groton. “I met her a week ago, and she wants me to define the relationship,” he said. “I’m technically single now. Technically.”  “Because you haven’t had that discussion yet?” Mr. Vest asked.  At 10:20 p.m., more than a dozen military men and women stood outside the Mean Fiddler, talking and smoking. The two sailors joined the group. Mr. Warren pulled on a vape and exhaled a great cloud of steam. “A lot more nicotine, a lot faster,” he said.  It was Jacob Underwood, 21, a fellow sailor who is from southeastern Oklahoma. After he embraced Mr. Vest and Mr. Warren, he said he hadn’t slept in two days. “By far the best bar has been the Mean Fiddler,” Seaman Underwood said. “It’s the sailor and Marine ‘come-to’ spot.”  The three men went inside. The Mean Fiddler was packed with sailors, Marines and Coast Guard members. “Shout-out to all the sexy ladies,” the D.J. hollered into a microphone.  Soon, four civilian women, two of them wearing Dixie cup caps, climbed aboard the bar and started dancing. When the song “Shots” by LMFAO (featuring the rapper Lil Jon) came on, the place went nuts. Sailors and Marines pushed their way to the bar, piling into one another like rugby players. Almost everybody sang along to the track: “Shots, shots, shots, shots!”  “What happens in the Fiddler stays in the Fiddler,” the D.J. shouted.  A civilian woman lay on the bar, supine. A bartender poured Kahlúa into the wide-open mouth of a Marine. The Marine bowed his head and spat the booze on the exposed space between the woman’s T-shirt and cutoff shorts. Then he slurped up as much of it as he could from her navel before helping her off the bar. They started to make out furiously.  “We’ll party like this all night long,” the D.J. said. “Do I have any military in the house?”  A lusty yell went up from the crowd. You could count the number of male civilians on one hand.  “This is Fleet Week,” Mr. Warren said. “This is what it’s all about.”  Mr. Vest and his two friends repaired to a less rowdy bar nearby, Dutch Fred’s, where the talk turned to the presidential campaign.  “I vote Trump,” Mr. Underwood said.  “I would have voted for Bernie,” Mr. Vest added, “but he’s pretty much out.”  “Trump’s the better option of the two,” Mr. Underwood said, “because the last two presidents we’ve had … ”  Mr. Vest cut him off, saying, “I like Obama.”  “Get out of here!” Mr. Underwood said.  “He’s our commander in chief,” Mr. Vest replied.  The conversation soon hit on other topics: national security, the presidency of George W. Bush, the possibility of another major war, the women at the Mean Fiddler. Mr. Underwood said he would like to go back there. But Mr. Vest seemed to have something else in mind.  “We’re missing ‘Game of Thrones,’” he said. “We already have missed it.”
Times Square. The Mean Fiddler. A slice at Two Boots. A 19-year-old sailor visits New York for the first time.