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.
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.