If you don't have access to a computer at home or at work, then Internet cafés are the next best thing. They generally aren't too expensive and can be easily found, especially in city centres.
Internet cafés have recently become so popular that many stores and calling centres have added a few terminals for their customers. However, the best places both in terms of speed and price tend to be the larger ones (20+ computers) that focus on providing computer access.
Cafés will charge between CHF1.00-1.50 an hour. Once you purchase access time, any time you have left over will not be refunded. If you expect to be online for a while, then it would be advisable to buy a special multi-hour pass. These passes usually come in increments of 5-10 hours and reduce the overall cost per hour. They also offer the advantages of multiple sessions and of only billing you for the minutes that you actually use.
Fees tend to only give you access to the computer and the Internet. Just about everything else might carry an extra charge: printing, scanning, and even copying files to and from floppy disks. Some larger cafés have begun billing additionally for the right to use spreadsheet or word processing programs (i.e. Microsoft Excel or Word).
Warning: Internet cafés are popular targets for thieves and pickpockets particularly in the centre of large cities. Keep an eye on your valuables at all times.