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. Costs vary a lot from £1.50-5.00. Fees tend to give you access to the computer and Internet. Just about everything else might carry an extra charge: e.g. printing, scanning, CD-Rom burning, etc.
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.