I wake up. Just. The headache is probably attributable to dehydration hastened by six hours of verse speaking followed by a late-night beverage or three sourced at the Dirty Duck. More on that later.

It's Sunday, my mother, sister, great-aunt and nine-year-old second cousin are all on their way to spend a day in and around the Bard's stomping ground. What in Oberon's Puckery is one to do?

Breakfast? A note here: actors on the whole don't do breakfast, and certainly not on a Sunday. Having placed the relatives in the Church Street Townhouse (16 Church Street, 01789 262222, churchstreettownhouse.com, doubles from £110) and the Legacy Falcon hotel (Chapel Street, 08444 119005, legacy-hotels.co.uk/legacy-thefalcon, doubles from £90), they'll have had a fantastic breakfast, so there is no need to panic.

Right. We're in Stratford-upon-Avon and I'm at the Royal Shakespeare Company for the first time in my career, so we should do something Shakespeare-y. We could head to one of the five famous houses maintained by the Shakespeare's Birthplace Trust (shakespeare.org.uk) adult five-property ticket £21.50, £13.50 for children). But it turns out they have already visited Shakespeare's birthplace (the Shakespeare Centre, Henley Street). The nine-year-old would definitely enjoy Mary Arden's Farm (Station Road, Wilmcote) with its falconry displays but that's three miles from Stratford-upon-Avon and I still have a headache. Aunt Maud would be delighted to wander through the award-winning gardens at Anne Hathaway's Cottage (Cottage Lane, Shottery), but it's winter.

I resolve to take them to my favourite: Nash's House and New Place (22 Chapel Street).

Shakespeare spent his final 19 years, and died, at New Place. He purchased this, the second largest of the trust's properties, in 1597. It's now an archeological site that has produced artefacts that are displayed next door in Nash's House on the corner of Chapel Lane. With its dark wood beams there is an authentic sensation of experiencing the past, especially as you creak up the surprisingly substantial central staircase.

Inside, displays explain the preservation of the building, Shakespeare's wealth, and his tax evasion (he was listed for it twice in London). They also reveal some interesting facts about famous Shakepeareans throughout history. For instance, Bram Stoker [the creator of Dracula] managed the Lyceum theatre in London for 20 years and oversaw Henry Irving's time there.

There's a sweet but small activity room, reminiscent of a parish creche, where my young cousin can design a 16th-century knot garden.

You can wander through to the Great Garden, a genuinely peaceful green space with a lawn surrounded by dense topiary, hosting several Greg Wyatt sculptures inspired by Shakespeare speeches. The garden's mulberry tree, with its gnarled trunk, was reputedly grown from a cutting of a tree the playwright himself planted. This is where to come to evade the RSC bustle and eat a mid-rehearsal sandwich.

Shakespeare done. What next? Several RSC members have been banging on about butterflies … A short walk across the marina, taking in the best view of the RST, and we're at the Stratford Butterfly Farm (Swan's Nest Lane, 01789 299288, butterflyfarm.co.uk, adults £6.25, children £5.25). A tropically warm greenhouse, it hosts jungle foliage, a large pond and modest waterfall arrangement – and thousands of butterflies that could not be more eager to land on your sleeve and dazzle you with their kaleidoscopic colours.

It's brilliant, a genuine reason to visit Stratford-upon-Avon in its own right. You can take in the enclosure and the caterpillar, arachnid and insect rooms in an hour or so, but I spend my time in there with a childlike gaze. I don't spy one of the three elusive iguanas but I spend an age staring at the worker ants carrying leaves along a 20ft rope down into the colony, and am charmed by the troupe of chubby flightless birds that decide to guide me around.

For a Sunday roast, we try the Church Street Townhouse hotel, an elegant yet homely restaurant housed within a delightfully illuminated 400-year-old Grade II-listed building. The food is fantastic (have the pork belly if they run out of roasts), the service attentive and friendly, and the atmosphere is enlivened on Sunday evenings, should you carry on, by local musicians playing acoustic music that floats through from the bar. A pleasing alternative can be found in the great food and unrivalled view from the Rooftop restaurant at the Royal Shakespeare Theatre (Waterside, 01789 403449, rsc-rooftop-restaurant.co.uk).

Child and great-aunt dispatched, it's time for a drink. Stratford-upon-Avon may not boast a vibrant nightlife but it does pubs very well. On a performance night, the cast of the show stop by the Swan theatre bar to pick up guests. It's a cosy snug of a study centred around a fireplace and situated within the RSC building. We'll have a relaxed drink and recline in "the insult chair" and later head off to the Black Swan pub on Waterside, aka the Dirty Duck (01789 297312, dirtyduck-pub-stratford-upon-avon.co.uk).

For generations RSC actors have imbibed, complained about rehearsals, competed to have their photos hung alongside [Judi] Dench and [Richard] Burton, and most importantly met members of the audience in this notorious tavern. Typically, most cram into the right-side bar: a narrow cove, no bigger than half a tube-train carriage, littered with photographs of actors. The air hangs thick with thespian (not luvvy) anecdotes and the nightly mingling of audience and company creates a circus-family atmosphere, transient yet warm. There is nothing more pleasing for an actor than to garner a reaction, and because it is a small and sleepy town, this post-show engagement is what makes acting in Stratford-upon-Avon so special.
• Gethin Anthony plays Grigory in the RSC production's of Boris Godunov, and also appears in A Life of Galileo at the Swan theatre (rsc.org.uk) until 30 March