Documentation on how to use Live Search API can be found here:
http://msdn.microsoft.com/en-us/library/dd251056.aspx. Since you are using ASP.NET I would recommend using either the SOAP interface or, if you are aiming for an AJAX implementation, our JSON interface.
As for Google, they only have a JSON interface, so you have a choice of AJAX or you can write a WCF REST client for their service. Try searching on
http://www.live.com/search?q=google+search+api for Google API resources.
Eric Carter - SDET II - Live Search API