Why go now?
This creative bayside city comes alive in December, at the start of the long summer. Melbourne Now, an ambitious arts festival, is currently under way (until 26 March) at the National Gallery of Victoria (1). The biggest exhibition in the venue's history, it provides an excellent overview of the cultural scene (00 61 3 8620 2222; ngv.vic.gov.au).
Throughout this month, the Christmas Festival (until 24 December) also offers a city-wide programme of family-friendly events (00 61 3 9658 9658; thatsmelbourne.com.au). Alternatively, beg, borrow or steal a ticket to the Boxing Day Test at the Melbourne Cricket Ground (2) in Yarra Park (00 61 3 9657 8888; mcg.org.au).
The only airline flying direct to Melbourne from the UK is Qantas (0845 774 7767; qantas.com), from Heathrow via Dubai. Connecting flights are available on a wide range of airlines. From Tullamarine airport, the SkyBus to Southern Cross Station (3) in the city centre departs every 15 minutes and costs A$17 (£9.70). The journey takes about 20 minutes. Alternatively, the Star Bus shuttle (half-hourly, 6am-6pm) will take you to your hotel for A$17-25 per person (£9.70-14).
Get your bearings
The grid-iron Central Business District (CBD) is at the heart of the city, on the Yarra river's north bank. The tourist office (4) is here, on the corner of Federation Square (00 61 3 9658 9658; visitvictoria.com; 9am-6pm daily). The gentrified eastern suburbs of Fitzroy, Richmond and South Yarra offer a plethora of shops, restaurants and cafés. Melbourne is often called Australia's "Garden City" thanks to its extensive parks, which run either side of St Kilda Road across the river to the south. Trendy St Kilda itself is located along the golden beach that lines Port Phillip Bay.
Trundling around the CBD on the City Circle tram is free, but for further afield, a Myki travelcard with one day's travel on it costs A$14 (£8) at rail stations or major bus or tram interchanges.
The bike-share scheme has 50 stations and costs A$2.80 (£1.60) a day to join – helmets compulsory.
The Hotel Windsor (5) at 111 Spring Street (00 61 3 9633 6000; thehotelwindsor.com.au) is opposite parliament and next door to the Princess Theatre. This grand old establishment opened in 1883. Couples should opt for a superior room (standard rooms have only queen-sized beds) from A$255 (£146), room only.
For a more contemporary feel, head to The Cullen (6) at 164 Commercial Road (00 61 3 9098 1555; artserieshotels.com.au), in fashionable Prahran which showcases work by Adam Cullen, Australian art's enfant terrible. Doubles start at A$189 (£105), room only.
There's lots of choice for tight budgets. Try Nomads Backpackers (7) at 198 A'Beckett Street, by Queen Victoria Market (00 61 3 9328 4383; nomadsworld.com). There's a sundeck on the roof and a cinema lounge. Doubles from A$140 (£80), bunks from A$29 (£16.60).
Take a view
The Eureka Skydeck (8) at 7 Riverside Quay (00 61 3 9693 8888; eurekaskydeck.com.au), has a 360-degree observation deck on the 88th floor and an incredible view to the Dandenong Mountains to the east. Those with steady nerves can step on "The Edge", a glass-floored viewing platform, to look at the street 300m below. The admission is A$18.50 (£10.60), but keep the ticket and return later in the day to see the stars in the night sky for only A$3.50 (£2) extra.
Get into the maze of lanes and arcades between Collins and Lonsdale streets. Madam Virtue & Co (9) at 5 Crossley Street (00 61 3 9663 8669; madamvirtue.com.au) offers one-off designer garments and haute couture. Closed Sunday.
Upstairs at 2 Somerset Place, Captains of Industry (10) (00 61 3 9670 4405; captainsofindustry.com.au) offers bespoke menswear and handmade shoes.
Lunch on the run
Take your pick of the al fresco cafés crowding Block Place, or try stylish Caffe e Torta (11) across the road at 314 Little Collins Street (00 61 4 3459 5885; caffeetorta.com), which is renowned for its coffee. The pumpkin frittata costs A$13.95 (£8).
Take a hike
Admire the street art on Union Lane (12) before heading north on to Bourke Street to check out the animated Christmas window displays at Myer department store (13), which have become a Melbourne institution. Follow Swanston Street north to the City Baths (14), then take a right along Victoria Street past Old Melbourne Gaol (15), an imposing bluestone building where bushranger Ned Kelly was hanged in 1880.
Wander the length of Carlton Gardens past Melbourne Museum (16) then follow Moor Street to the foot of Fitzroy's thriving Brunswick Street, known for its live-music venues and vintage fashion stores. Take a left, then left again after six blocks to find Rose Street Artists' Market (17) at 60 Rose Street (00 61 3 9419 5529; rosestmarket.com.au; weekends, 11am-5pm) where emerging artists and designers sell their wares.
Sit streetside or cuddle up in a love seat at the Black Cat (18) at 252 Brunswick Street (00 61 3 9419 6230; blackcatfitzroy.com), a popular café/bar with a 1970s vibe, leafy beer garden and Monteith's cider on tap. DJs take to the decks towards the end of the week.
Dining with the locals
A few doors down, Naked in the Sky (19) at 285 Brunswick Street (00 61 3 9416 2238; nakedforsatan.com.au) serves peasant-style Basque food with spectacular views over the city from its rooftop restaurant. The grilled octopus with potato purée costs A$26 (£15). Open late, no reservations.
Alternatively, head back into town to Cumulus Inc (20) at 45 Flinders Lane (00 61 3 9650 1445; cumulusinc.com.au) for fine wine and unpretentious food in this airy, minimal eatery. Try to get a seat at the marble bar overlooking the kitchen. Open daily until 11pm.
Sunday morning: go to church
The neo-Gothic St Paul's Cathedral (21) is on Flinders Street opposite Federation Square (00 61 3 9653 4333; www.stpaulscathedral.org.au). Highlights include the intricate tiled floors and distinctive banded stonework. Choral Eucharist starts 10.30am on Sundays.
Out to brunch
Melburnians take brunch very seriously. Every restaurant worth its salt will open early at the weekend with an offering at least as creative and exciting as its night-time menu. Linger over baked eggs with chorizo (A$18.50/£10.60) or rice pudding and rhubarb (A$10.50/£6) while sitting in the sunshine outside The European (22) at 161 Spring Street (00 61 3 9654 0811; theeuropean.com.au).
Walk in the park
From Spring Street, follow Treasury Lane past the government buildings to find lush Fitzroy Gardens (free entry). The fuchsia display in the Conservatory (23) is at its peak at this time of year. Cook's Cottage (24) near Wellington Parade (00 61 3 9419 5766; www.cookscottage.com.au; 9am-5pm; A$3.50/£1.80) was the childhood home of explorer James Cook, built in Yorkshire then reconstructed brick by brick here in the 1930s.
Take a ride
Cycle along the shores of Port Phillip Bay on the Bayside Trail from Cheetham Wetlands in the west to Curram to the south-east. Join the track at Station Pier (25) in Port Melbourne and follow it down the beach for 5.5km to St Kilda, then break for a dip in the Sea Baths (26) on the foreshore (00 61 3 9525 4888; stkildaseabaths.com.au). Tram 16 will drop you back in town at Flinders Street station (27).
The Ian Potter Centre (28) on Federation Square (00 61 3 8620 222; ngv.vic.gov.au; 10am-5pm daily except Mondays; free) is the National Gallery of Victoria's second site. It's devoted to Australian art, from the colonial period to the present day, and includes work both from indigenous and non-indigenous artists.
Icing on the cake
Open-air cinemas are all the rage. The Moonlight Cinema at the Royal Botanic Gardens (29) in South Yarra, has been running the longest. Gates open at 7pm, screening starts at sundown – until 30 March (moonlight.com.au).
Or try the Shadow Electric in the beautiful grounds of Abbotsford Convent (30) on St Heliers Street, for highbrow fare until 24 January (shadowelectric.com.au).