The cast of “Kiss Me, Kate” performs at the Shakespeare Theatre Company. (Scott Suchman)

“Kiss Me, Kate” is a lively, lusty musical comedy, and the Shakespeare Theatre Company’s big, new staging takes a very fine straight-arrow approach. Director Alan Paul aims to get out of it what composer Cole Porter and book writers Samuel and Bella Spewack put in back in 1948: brawling romances, great tunes, low jokes and fancy dances.

The show makes its brassy, sweeping intentions clear from the start. A dusky saxophone introduces Porter’s theater anthem “Another Op’nin’, Another Show,” and the wide stage at Sidney Harman Hall gradually fills with sexy showbiz types. Choreographer Michele Lynch knows just what to do with them. Jazzed by hormones and adrenaline, these characters — actors on an out-of-town “Taming of the Shrew” tryout in Baltimore, if you recall the plot — naturally cluster into flirty little twosomes and then into grand patterns that pop with joy.

You can see that Paul is not out to reinvent the wheel; he just wants to see if he can make it spin like a top. James Noone’s spiffy set design has fun with wobbly 1940s-style scenery while giving Lynch’s dancers staircases to hop up and surprising props to bounce off. Costume designer Alejo Vietti’s filmy period suits, skirts and shorts are fetchingly designed for these romantically combustible characters to preen in. Music director Doug Peck’s 10-piece orchestra motors the Porter score from Renaissance harpsichord accompaniment to jazz-band swing.

That’s the canvas for the “Taming of the Shrew” story of Fred Graham and Lilli Vanessi, the divorced stage stars who still can’t keep their hands off each other, for better and for worse. (The Spewacks’ book was updated for the 1999 Broadway revival by an uncredited John Guare.) Caresses are followed by slaps; kisses lead to punches. As the outsize Fred and Lilli, Douglas Sills and Christine Sherrill create a pair of equally matched egomaniacs. The Sills-Sherrill roaring may be more persuasive than the wooing, but not by much, and both actors play the comedy with a welcome touch of class.

Sills is a real wild card. He’s a leading man with great diction, a solid singing voice and the spirit of a clown; he’s a strapping figure who relishes physical comedy and dry zingers, which makes him an excellent fit as the bossy, hammy Fred. If Sherrill seems remote at first, she begins to shed it with the dusky solo “So in Love,” the great Porter torch song. She is a formidable Lilli (as she was a formidable evil political mother recently in Signature Theatre’s “The Fix”), and she soars impressively through the comic rage of “I Hate Men.”

The second-banana romance is as combative, with foxy club star Lois Lane (playing Bianca in “Shrew”) moaning the song “Why Can’t You Behave?” to the rover Bill Calhoun. Clyde Alves is a springy, disciplined dancer as Bill, bounding and pirouetting through the comic love tune “Bianca.” (Lynch’s accomplished, fresh choreography really catapults the show up a level; first-rate dancing is still the rarest thing for locally produced musicals to pull off.) As the buxom Lois, Robyn Hurder has plenty of chances to show that she can dance, too; she’s the centerpiece of the tap-dancing quartet “Tom, Dick or Harry” with Alves, Brandon Bieber and Con O’Shea-Creal, and every wiggle, spin and high kick in her sassily sung solo “Always True to You in My Fashion” is polished to a gleam.

“Kiss Me, Kate” is vintage Broadway-style entertainment — this is the musical with the sultry, explosive dance number “Too Darn Hot” and the vaudeville “Brush Up Your Shakespeare,” both neatly handled here — and Paul continues to show that he knows how to snap together such a massive puzzle. Big musicals have been rolling in on Washington’s large stages all month — “Oliver!” at Arena Stage; “Rodgers and Hammerstein’s Cinderella” at the National Theatre; “Guys and Dolls” at the Olney Theatre Center — and wait! More are afoot before Christmas, including Broadway’s “Matilda”; the new Steve Martin-Edie Brickell project “Bright Star” at the Kennedy Center; and Signature Theatre’s upcoming “West Side Story.” If there’s a knock on “Kate,” it’s that even its well-judged components feel mechanical here and there — but just a touch. The STC has been at the musicals game for about six minutes; Paul and company are stretching out and learning fast.

Kiss Me, Kate Music and lyrics by Cole Porter, book by Samuel and Bella Spewack. Directed by Alan Paul. Lights, Paul Miller; sound design, Justin Stasiw. With Bob Ari, Raymond Jaramillo McLeod, Patrick Ryan Sullivan, T. Oliver Reid, Zonya Love, Bev Appleton, Harry A. Winter and Elliot Dash. About 2 1/2 hours. Through Jan. 3 at Sidney Harman Hall, 610 F St. NW. Tickets $20-$118, subject to change. Call 202-547-1122 or visit www.shakespearetheatre.org.