There is a growing variety of online ways of finding a climbing partner, including some apps.
Climb Buddy (app) - An app for iPhone and Android. People have had different amounts of success with this app, think it only really works in major towns and cities where there is more people. Has good filtering features and a good concept.
Facebook - There is a lot of groups on Facebook now, covering the majority of all climbing areas in the UK. I have used these groups a few times and one person I still climb with occasionally now, great way to meet new people.
These can be interesting. Most are filled with the older generation, with little, if any younger members and usually require you to pay a annual membership fee. It's a great way however of getting to meet loads of like minded people, especially if just starting out as members will usually be keen to help you out.
Either just hover around and talk to people, introduce yourself and ask if you can climb with them, we're generally a friendly bunch! Most walls now usually have a social group or evening where you can go along to meet new people.