Many of the internet cafes that have emerged in the 1990's have closed since, probably because Switzerland has one of the highest rate of high-speed internet connections in homes in the world, but almost any video rental shop and most train stations will have a few internet terminals. The tourist office should be able to direct you to the nearest one. The going rate is 5 CHF for 20 minutes.
Also, you can send email, SMS (text messages to cell phones) or short text faxes from just about every public phone booth for less that 1 CHF. Some public phone booths allow you to browse the internet. There are many shopping centers and cities (Lausanne and Vevey for example) that offer free wireless internet access: ask the young locals, maybe they know where to go.
The public phones are surprisingly cheap, and have no surcharge for credit cards.
If you stay for some time, it may be advisable to buy a pre-paid cell phone card that you can use in any phone that supports the GSM standard on the 900/1800 Mhz bands - they usually cost around 10-40 CHF and are obtainable in the shops of the mobile service providers Swisscom, Orange or Sunrise in most cities. Mobile network coverage is close to 100% by area, even in the mountainous, non-populated areas.
There are also a lot of cheap prepaid cards for local calls from other providers. The prepaid cards of the big supermarket chains Migros (M-Budget-Mobile and Coop ( Coop Mobile for example cost around 20 CHF and include already 15 CHF airtime. The cheapest prepaid card for calls within Switzerland is Aldi Mobile 0,14 CHF/min Switzerland fixed and Aldi mobile, 0,34 CHF/min other mobiles. The cheapest prepaid card for international communication is yallo 0,39 CHF/min within Switzerland as well as to all European and many more countries (to the mobile and fixed networks). This includes the UK, USA, Canada, Australia and New Zealand. SMS cost 0,10 CHF. The prepaid cards can be bought online (30 CHF with 30 CHF airtime inclusive), in most post offices (29 CHF with 20 CHF airtime inclusive) or Sunrise shops (20 CHF with 20 CHF airtime inclusive). An other prepaid card with cheap rates offers Lebara Mobile (Sister concern of Sunrise). The prepaid card is available for 5 CHF with an equivalent talk time and recharge vouchers offer the talktime equivalent to the price of the voucher.