1. Which languages do you know? Or want to learn anyway? You may well have a few learning curves to ascend apart from just a server language.
2. Everything that runs on the web server and generates web pages should work with any decent browser - you'll have to test your site on a variety of browsers anyway if you want it to work on whatever platform your users happen to have (any site should be happy with Mac / Windows / Linux as clients, with recent versions of Internet Explorer / Firefox / Opera / Safari / Chrome) - and even more testing if you want it to work on a reasonable variety of mobile devices.
4. Server side - ASP.NET (using C# or VB) and PHP are fairly common - but there are plenty of others.
5. Depending on complexity, you might need a Content Management System (e.g. JoomLa, which is free).
6. You may also need a database to support whatever you do (MySQL / SQL Server are common) - and may need to know a bit of SQL to use it depending on what server language / other facilities you use.
7. What you go with may also depend on whether your hosting is on a Windows or Linux (or other) server / what features your hosting provider offers / what they charge.
8. For any of the above you may have to learn how to configure things on your server, get components to work together, get to grips with a development environment . . .
9. To integrate with Facebook you'll have to get to know the FB API and how to generate the relevant web requests / handle the responses
10. Speed - on a decent commerical web hosting service, all of these are fast - with the caveat that if the coding / design of a site is sufficiently inefficient anything can be made to go as slowly as required :-) Seriously, if it isn't as fast as you want, (a) find out what is slowing it down and/or (b) pay to upgrade to a beefier server platform - and speed is partly about how many concurrent requests can be handled without a noticeable slowdown as well as raw speed for a single request which, beyond a certain point, may become less important.
As a general resources, W3Schools (http://www.w3schools.com/
]) is a good starting point.
Personal preference : Windows hosting with ASP.NET / C# and SQL Server database - which should deliver everything you've specified
Second choice : PHP/MySQL/JoomLa - which would probably also deliver everything you've specified, though I haven't personally tried all the relevant bits (e.g. AJAX or accessing FB) so you'd need to do a bit more research.
I use fast2host.com and have been well pleased with the combination of facilities / cost and good support.
That probably goes beyond the strict remit of your question, but I don't think you can consider what language you use properly without considering all the other stuff that goes round it.