To get current user info:
jQuery.ajax({
url: _spPageContextInfo.webServerRelativeUrl + "/_api/web/currentuser",
type: "GET",
headers: { "Accept": "application/json;odata=verbose" }
}).done(function( data ){
console.log( data );
console.log( data.d.Title );
}).fail(function(){
console.log( failed );
});