To get internet access wirelessly, you either need a wireless router (or combined router/modem) in your home connected to your broadband, or connect to a wifi hotspot that can be free in places like McDonald's, some libraries and cafes; or paid for via OpenZone or The Cloud
http://www.btopenzone.com/
http://www.thecloud.net/for-you
All of these options aren't truly mobile - you have to be within range of a wifi hotspot.
Or you can buy a dongle from Vodaphone; 3, TMobile, O2 or BT which gives much greater coverage on a contract or pay as you go option.
As the rates and offers change daily it is impossible to give advice re best prices without spending time researching it - which you can do just as well as me.
Be aware that not all areas of the UK have access to mobile internet because the coverage is not good; it can be slow and unsatisfactory and in places will not work at all.
Also, if you are contracted to a specified amount of useage, it can be horrendously expensive if you exceed your limit