The city is defined by Lake Geneva, whose western end it wraps around. The Jet d'Eau is an icon for the city. The 140-metre fountain is more than a tourist sight – it began as an imaginative way to relieve excess pressure in the city's water-management system. Seven tonnes of water are airborne at a time.
Just 400 metres to the west, the replenished Rhône flows out of the lake on its way to the Mediterranean, and is crossed by six bridges – starting with the busy Pont du Mont-Blanc, then a pedestrian link to Ile Rousseau, with its statue of the philosopher, and the Pont de la Machine. The main tourist office is north of the river (on the right bank) at 18 Rue du Mont-Blanc (www.geneve-tourisme.ch; 9am-6pm daily, Monday from 10am, Sunday 10am-4pm). It sells the Geneva Pass, granting access to sightseeing tours, museums and other attractions (24 hours is Sfr25/£19; 48hrs, Sfr35/£25; or 72hrs, Sfr45/£32).
Hotels also provide a free "Unireso" card giving unlimited use of city buses, trains, trams and boats.
Take a hike
From the tourist office, walk towards the lake and cross the Rhône over the Pont du Mont-Blanc. Wander through the 1854 Jardin Anglais and get a time check from the impressive flower clock.
Go one block inland and right along rue du Rhône to the elegant Tour du Molard, marking the start of the broad Place du Molard. It contains a relief inscribed "Genève Cité de Refuge"; search the cobbles for one-word messages such as "Gracias" and "Bienvenue". Then fix your sights on the towers of the cathedral and climb a series of staircases to the old town.
On Rue du Puits-Saint-Pierre, call in at the concise Maison Tavel at No 6 (mah.ville-ge.ch; 11am-6pm Tuesday-Sunday), which tells the story of Geneva through the centuries.
The Hotel de Ville was where the first Geneva Convention was signed in 1864, the initial act of the International Red Cross.
Take a brief detour to the Promenade de la Treille, an elevated terrace that looks out towards the Jura Mountains and claims the world's longest wooden bench, at 126 metres.
Take a view
The Cathédrale St-Pierre (saintpierre-geneve.ch; 10am-5.30pm daily, Sundays from noon) offers the finest view from its twin towers. Buy a Sfr5 (£3.50) ticket from the desk and climb through the innards until you emerge high above the Old Town. Each tower offers a different glimpse of the city. Back at ground level, explore the cathedral – which is mostly a good definition of the word "austere".
It was here that John Calvin preached reform in the 16th century, and his uncompromising chair is prominently displayed. To imagine how the whole cathedral must once have looked before its adornments were stripped away, visit the colourful Chapelle des Macchabées.
Lunch on the run
You are a few steps from the Place du Bourg-de-Four, site of the old Roman forum. Today, it has the feel of a village square rather than the kernel of a world city. The best pit-stop is Chez Ma Cousine at number 6 (chezmacousine.ch), offering free-range chicken dishes and salads 11am-11.30pm daily.
Given the strength of the Swiss franc, you may not want to go much further than merely looking – but for bargains, visit the flea market that spills across the Plaine de Plainpalais on Saturdays. Close by, Le Globe-Trotter at Bld des Philosophes 9 (leglobetrotter.ch; 10am-4pm Saturdays; 10am-6pm Tuesday-Friday) dispenses active and outdoors gear and travel goodies galore.
The main island in the Rhône has a bright and characterful bar at its western end in what used to be a wholesale market: the Brasserie des Halles de l'Ile (brasseriedeshallesdelile.ch). Note the minimum Sfr20 (£14) spend – you could reach the target by eating here.
Dining with the locals
Alternatively, aim north of the Rhône to the indulgent, richly decorated Café de Paris at Rue du Mont-Blanc 26 (chezboubier.com; 11am-11pm daily). The only dish at this theatrical bistro is sirloin, cooked to your wishes and served with herb butter, three helpings of frites and green salad. Simple, but heavenly; Sfr42 (£30).
Go to church
The onion domes of the Russian Church at Rue de Beaumont 18 tower above the south east of the city. It was built in 1863 by Queen Victoria's aunt, Princess Juliane of Saxe-Coburg-Saalfeld; she married into the Russian aristocracy but preferred life in Geneva to the Imperial court in St Petersburg. Wander in and around to admire the wealth of icons.
Out to brunch
The machine that gave the Pont de la Machine its name formerly occupied the building in the middle and pumped water to the city's lesser fountains. Today, the structure houses the Cité du Temps – an unusual combination of art space, Swatch museum and chic lounge-restaurant, Au Phil du Temps (citedutemps.com). It opens 9am-6pm daily.
Walk in the park
The Parc des Bastions offers a charming combination of formal greenery and outdoor chess. The main attraction, though, is the Mur de la Réformation – a 100-metre relief paying tribute to 16th-century Protestant reformers including Calvin and his ally John Knox, founder of Scottish Presbyterianism.
Walk a block west and take tram 15 to its terminus at Palais des Nations – home to the United Nations' main European presence.
To the left of the mural preaching peace, follow the road around to the Musée Ariana, a beautiful venue for ceramics and glassware (ville-ge.ch/ariana; 10am-5pm Sundays, to 6pm other days, closed Mondays).
On the other side of the road is the International Red Cross Museum at avenue de la Paix 17 (micr.org; 10am-5pm daily except Monday; Sfr15/£11). It tells the story of how the Red Cross/Red Crescent was founded by a Swiss businessman, Jean Henri Dunant.
Icing on the cake
As the sun dwindles over Switzerland's westernmost city, view it from the water with a trip aboard one of the mouettes that shuttle back and forth between the two sides of the city. The journey is covered by the Unireso Geneva transport card.