The Kenneth Branagh Theatre Company launches its year-long "Plays at the Garrick" Season with Shakespeare and Terence Rattigan playing in tandem through to mid-January: three hours of The Winter's Tale, with Branagh as Leontes and Judi Dench as Paulina, is paired (on a separate bill) with Rattigan's 1948 backstage farce, Harlequinade, and, as a bonne bouche, a short 1968 monologue, delivered by Zoë Wanamaker, All On Her Own.

The first thing to say about Saturday's opening (Shakespeare at the matinée, Rattigan by night) is that this is a generous, full-hearted, thoroughly enjoyable occasion, from the moment Jessie Buckley prefigures the heart-stopping beauty of the awakening in Shakespeare's Sicily with a plaintive melody in a light fall of snow right through to Branagh in tights and dodgy wig excoriating the Arts Council as an over-age Romeo on the rack.

The second thing? How wonderful that an actor of Branagh's sly brilliance and versatility is leading his own company on the West End stage, in a crowd-pleasing repertoire that blows away any mutterings of "old-fashioned" with its indisputable energy and gusto. With all due respect to Kevin Spacey at the Old Vic, we never got a sense of a great actor at the helm and the heart of his own operation, as we did with Olivier at the National or, fleetingly, Ian McKellen at the Actors' Company before he threw in his lot with the RSC.

Leontes is not an obvious Branagh role. But I remember thinking the same of his Coriolanus at Chichester, and he surprised me there, too. His Volumnia was Judi Dench, and Paulina is a similar sort of intervening character; Dench brings all her deeply felt wisdom and humanity to bear on the role. She's so good at disapproval, and when Hermione is falsely arraigned by her husband, she leaves no-one in two minds as to what she thinks, fending off attention and offers of help with a brusque, bustling exit that almost elicits a round on its own.

This Winter's Tale, loosely Edwardian in Christopher Oram's simple, effective palace design of pillars and vistas, starts at Christmas and freezes over in the slowly calibrated decline from warmth and hospitality to jealousy and catastrophic consequences. Branagh is an old school director/performer (he shares directing duties with Rob Ashford) in that he's not afraid to chart Leontes' changing mood and temper with the underpinning of Patrick Doyle's music or the focus of Neil Austin's lighting. His wife Hermione (a beautiful, perfectly sculpted performance by Miranda Raison) is "sluiced" and "slippery" – Branagh detonates these expressions of disgust with cold ferocity – while "paddling palms" with Hadley Fraser's attractive visitor, his best friend, Polixenes.

The play is one of emotional and geographical extremes: the passage of time denoted in Dench's riveting Chorus at the start of the fourth act; the dilemma of Michael Pennington's dear old Antigonus abandoning the baby on the beach before the famous "exit pursued by a bear" (I've seen that stage direction better done, though); the raucous scenes of shepherds, sheep, flowers and John Dagleish's high tenor Autolycus, that "snapper up of unconsidered trifles," leading the folksy fandango in the suddenly softer world of a bamboo-bedecked Bohemia.

You won't see a better version in terms of heart, bones and lucidity. Rattigan's Harlequinade – more usually paired with The Browning Version – is equally hard to pull off, but Branagh as the ludicrous juve Arthur Gosport, Zoë Wanamaker as the bibulous Nurse, Dame Maud, and John Shrapnel as a bluff stalwart, George Chudleigh, get the tone of jobbing, faded theatricality just right. Both shows are lit up by Jessie Buckley, Tom Bateman, Jimmy Yuill and Vera Chok.

Perhaps surprisingly, Dame Judi, who's not in the Rattigan, doesn't deliver the short curtain-raising monologue All On Her Own, first written for Margaret Leighton on television, once performed by Barbara Jefford at the King's Head; but Wanamaker is devastatingly good as a bereaved woman, Rosemary Hodge, trying to convince herself that her husband hasn't committed suicide.

The piece touches, too, on a kind of melancholic sexual regret and self-laceration distinctive and peculiar to Rattigan and, at just fifteen minutes, is like one of those short Alan Bennett Talking Heads, only shorter, and balder.

The Winter's Tale and Harlequinade run at the Garrick Theatre until 16 January 2016.