Taylor and Harry's PDA at friend's birthday party

Taylor is wearing Harry's Paper Airplane necklace. She first stepped out wearing the necklace in early November

Their romance is heating up! New couple Taylor Swift and One Direction's Harry Styles stepped out at New York's Crosby Hotel on Thursday night to celebrate a friend's birthday – and get in a little PDA of their own. 

Swift first met up with friends at the McKittrick Hotel, where they hit up Manderley bar and then caught a production of the interactive theater experience Sleep No More. After the show, they moved the party to the Crosby Street Hotel, where they were joined by Swift's new beau. Along with Dianna Agron, Emma Stone, Jessica Stam and about a dozen guests, Swift, 22, dressed demurely in black, and Styles, 18, watched as a cake was served and helped to sing along to "Happy Birthday".

"Taylor and Harry [were] being very smoochie," a source at the party told PEOPLE. "They definitely [looked] like a couple." 

According to the source, Swift (who is up for three Grammys) and Styles held hands and he seemed "really protective of her" and kept his arm around her during the party. They shared plenty of kisses, too, and didn't seem to mind if anyone saw. Says the source: "They're not trying to hide it. The bar was packed." 

Their outing marks a string of get-togethers in New York City over the past week. On Sunday they visited Central Park Zoo together. And on Monday, Swift attended a Madison Square Garden show afterparty at the Hudson Hotel for Styles' band, One Direction, where the couple joined for a karaoke duet of a Backstreet Boys song. Swift later sang a One Direction tune, before the couple returned to her hotel around 4 a.m. 

The couple began quietly dating at the start of October, after "reconnecting" at September's MTV VMAs. (They first exchanged numbers at the Kids' Choice Awards in March.) The pair engaged in "serious flirting" at London's Teen Awards in early October and were seen holding hands at the X Factor studios in LA the following month.

Emma, Dianna and Jessica at dinner
People.com pictures