Having scoured the Internet via search, Facebook and Twitters own documentation, I am at a loss as to which APIs I need to use for my purposes.
I have a site which organisations can register to, and these organisations can add events. The public can browse through the organisations or the events. When either type is registered, I want to ask the user to provide a facebook and/or twitter URI (or just username/ID, whatever is required) so that I can display the latest 10 posts to the timeline or tweets.
I believe that this is possible, which APIs do I need to use? Where is some documentation explaining how to do this?
You will need to use each platform's API. Note that you will probably want to use Twitter via Javascript because Twitter has a pretty low volume API per IP address. You will want to push that to the client for a diverse IP pool.
https://dev.twitter.com/docs/api/1/get/statuses/home_timeline
here is how I addressed it:
http://blog.voltampmedia.com/2011/08/15/twitter-api-restrictions-and-ajax-to-the-rescue/
Facebook will allow you to request information repeatedly, so pushing to the client isn't required. Although you may want to consider loading via ajax so you are not waiting for the Facebook request to load your page.
If you are using java you can use spring social api. http://www.springsource.org/spring-social. It internally calls facebook graph api functions and you don't have to worry about parsing json data, it will do it for you.