Most international visitors will make a stop in Luzern. The city offers the old Kapellbrücke as well as old buildings, murals, houses, souvenirs shops, a stone lion and lots more. Just around the corner are the Vierwaldtstättersee (aka Lake Lucerne) and the mountains with Rigi, Pilatus and Titlis the best known.
Basel is a city on the River Rhine, bordering France and Germany. It is Switzerland's third-most-populous city and home of the pharmaceutical industry. Basel is commonly considered to be the cultural capital of Switzerland with a variety of museums and the zoo.
Zürich is the financial capital and largest city of Switzerland. You'll find the head quarters of major banks, insurances etc here. Tourists may like the old city, the lake and summer festivals. The "Gold Coast" hosts several international celebrities.
Geneva is probably the most international city in Switzerland: several global organisations (e.g. United Nations, World Trade Organisation & Red Cross) and corporations have their head quarters here. For tourists the Jet d´Eau ("water fountain"), the many quality watch stores, the large Lake and the vicinity to France are most obvious.
Locarno is located in the Italian part of Switzerland and best known for its film festival. But there is more than that. The Lago Maggiore offers many water activities and neighbouring cities like Ascona and Bellinzona are also worth your visit.
Montreux is located at the east end of Lake Geneva and just West of the Bernese Alps. Chillon Castle and Charles Chaplin's museum can be found here. Once a year there is the world famous jazz festival. Yes, there is lots to do.