Just north of Italy with cantons that speak French, Italian, and German, although the native tongue of many in the latter group is "Swiss German," a language similar to but not mutually intelligible with standard German. However, as with most tourist destinations, many speak at least some English. Famous for skiing in the winter, Switzerland is exceptionally beautiful for hikers in the summer.
A city on the Northern corner of Switzerland that borders France and Germany. The citizens of Basel are genuinely friendly and are happy to help tourists (in English). The city is clean, virtually crime free, has extensive and efficient public transit (mostly light rail), excellent food, and is considered one of the cultural centers of Switzerland. Stay in the old town (near Marktplatz) to be in the middle of everything (almost every hotel includes free use of the transit system with your stay, but the city is easily walkable). There are restaurants all over the city, but the night life is certainly near the University on Steinenvorstadt (take the tram to Barfusserplatz) with a dozen or so bars open late. See the Munster Cathedral on the Rhein built in 1019-1500, and one of the three Roman gates (Spalentor, St. Alban Tor, and St Johanns Tor) still standing from the protective walls built around the city around 1400. There are numerous museums, as well, dedicated to art and music. Swiss are known for chocolate, but if you want to take it to a new level, try the Luxembergli or Pralines. Beware, though, that Basel can be very expensive and everyday items and food may cost 2-3 times what they cost in the US.
Around Basel are the cantons of Basel-Stadt, Basel-Landschaft, Aargau, and Solothurn.
Geneva, Vaud/Lausanne, Jura, Fribourg, and Neuchâtel
This is the (mostly) French-speaking western part of Switzerland ("la Romandie"), along with Valais (below). (Swiss people humorously refer to the line between the French west and the German east as the "Röstigraben", or "rösti ditch", since rösti is a dish very popular with German-speaking Swiss but not Romands.)
Is the leading city of the French-speaking region, on the shores of Lake Geneva ("Lac Léman" in French). Home to many international organizations. A bit boring.
Canton along the north end of Lake Geneva. Its largest town, Lausanne, is home to the International Olympic Committee. Lavaux is home to some very old vineyards.
Montreux - terraced town on the eastern end of Lake Geneva (Lac Léman), home to a jazz festival
Chillon Castle (Château de Chillon), near Montreux, is one most iconic sights in this region
Also called Freiburg in German (the canton is officially bilingual), although the town of Fribourg is mostly French-speaking.
Gruyères - medieval town with a castle, namesake of Gruyère cheese
Lucerne and Central Switzerland
The heart of German-speaking Switzerland, this central district hosts the town of Lucerne (Luzern), Lake Lucerne, and four surrounding cantons (Uri, Schwyz, Nidwalden, Obwalden)
Engelberg - popular mountain resort town (population ~4000), good skiing in winter and hiking in summer
Melchsee-Frutt - car free mountain resort village, near Lake Melchsee
Mostelberg - car-free village, accessible from Sattel, known for its crazy pedestrian suspension bridge
Rigi Kaltbad - car-free village on Mount Rigi near Lucerne, accessible via train from Vitznau or via cable car from Weggis
Stoos - car-free village of ~100, on the Fronalpstock
Home of beautiful hikes such as the Klingenstock – Fronalpstock panorama hike that begins and ends in the town of Stoos.
You can also take trains/gondolas (or hike) to the top of nearby Mount Pilatus and Mount Rigi.
Bern is the Bundesstadt or, capitol, of Switzerland. The city symbol is the brown bear (baren) and it is featured prominently on the city's crest and signage throughout the area. The Barengraben on the edge of the city center has been a bear park since the 16th century, where it is possible to feed carrots to the city mascots. It is notable that Julius Caesar mentioned Bern as an "oppida" (or, pre-Roman, Celtic Iron Age settlement) that he encountered during his conquests of Gaul.
The city center is a UNESCO World Heritage site and home to a 15th century, Gothic cathedral, the Munster. It's most famous attraction is likely the Zytglogge, which is a medieval clock tower that bears, possibly, the largest clock face in Europe. Another building of significance is the apartment where Einstein worked out his Theory of Relativity. Located on the Kramgasse a few meters from the Zytglogge, it is possible to tour the apartment for a fee.
While the city center of Bern is one of the best shopping districts in central Europe, do not attempt to shop on Sundays as most everything is closed.
The city is mostly German-speaking, but the surrounding canton is officially bilingual French/German (although there are a minority of French speakers).
***Lötschberg AOC - great, informal restaurant with lots of young people, lots of beers & wines, classic Swiss dishes (fondue, raclette, rösti), really pays attention to ingredients (as the "AOC" name implies), excellent rösti with uncooked ham and cheese. Menu is in German but staff all speak great English. Also can get items to take away.
Check out the Bernese Oberland page for more on this beautiful Alpine region south of Bern.
The mostly French-speaking (but also German-speaking, especially in the eastern half) Alpine region in the southwest has Sion as its capital and Zermatt as its most famous ski resort (just north of the Matterhorn, perhaps the most famous mountain in Switzerland, which also borders Valle d'Aosta in Italy...in Italian it's called Monte Cervino).
Belalp - car-free village, accessed via cable car from Blatten
Bettmeralp - car free village of ~400, accessed via cable car from Betten; another cable car goes almost to the top of Bettmerhorn
Riederalp - car-free village, accessed via gondola or tramway from Morel
Fiescheralp - highest of the car-free villages on the Eggishorn; accessible from Fiesch via tramway; another tramway goes to Eggishorn's Fiescherhorli summit
Gspon - car-free village accessible from Staldenried, in the Saastal
Rosswald - car-free village accessible from Ried-Brig via cable car
Saas-Fee - car-free village of ~1600
Zermatt - the most famous resort town in the region; no combustion-engine cars are allowed in town but there are electric cars. Cable cars lead to the Klein Matterhorn.
Northeastern Switzerland ("Ostschweiz", the cantons of Thurgau, Appenzell, St. Gallen, and Glarus, is a region best known for the Rhine Falls [and the nearby cute town of Stein-am-Rhein], whose largest town is St. Gallen) and Graubünden (a canton bordering Austria, known for ski resorts Davos, Klosters, and St. Moritz) in the southeast. Graubünden (Grisons) is mostly German-speaking, but the south is Italian-speaking and pockets speak Romansch (Switzerland's fourth national language, a Romance language somewhat related to the native dialects of Friuli).
Appenzell - rec'd by Rick Steves
Braunwald - car-free town of 300, accessible via funicular from Linthal, skiing & hiking, alpine rose gardens
Müstair - tiny Romansch-speaking town home to the early Medieval Abbey of Saint John, an UNESCO World Heritage Site
St. Gallen - largest town in Eastern Switzerland; its Cathedral of St Gall is an UNESCO World Heritage Site
Hinterrugg - highest peak in the Appenzell alps (ont he Churfirsten range); you can get to the peak via the Chäserrugg cable car
Check out the Zurich page for notes on Switzerland's largest city.