Page redirect with successful Ajax request

Sure. Just put something at the the end of your success function like:

if(result === "no_errors") location.href = "http://www.example.com/ThankYou.html"

where your server returns the response no_errors when there are no errors present.

Just do some error checking, and if everything passes then set window.location to redirect the user to a different page.

$.ajax({    url: 'mail3.php',    type: 'POST',    data: 'contactName=' + name + '&contactEmail=' + email + '&spam=' + spam,    success: function(result) {        //console.log(result);        $('#results,#errors').remove();        $('#contactWrapper').append('<p id="results">' + result + '</p>');        $('#loading').fadeOut(500, function() {            $(this).remove();        });        if ( /*no errors*/ ) {            window.location='thank-you.html'        }    }});

You can just redirect in your success handler, like this:

window.location.href = "thankyou.php";

Or since you're displaying results, wait a few seconds, for example this would wait 2 seconds:

setTimeout(function() {  window.location.href = "thankyou.php";}, 2000);