The best places both in terms of speed and price tend to be the larger Internet Cafés (20+ computers) that focus on providing computer access.
Cafés charge €1-2.50/hour. Once you purchase access time, time you have left over will not be refunded. If you expect to be online for a while, it's advisable to buy a special multi-hour pass. These passes usually come in 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 (e.g. Microsoft Excel and 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.