PHP, JQUERY, AJAX mailing form

I have a simple mailing form on my website that works in the old fashion way meaning that all inputs are collected and send to sendmail.php that handles mailing of the form and also shows the result to the user informing that the form was submitted to us. All of it works fine, but the only problem I have is that user after clicking submit is transferred to another page (sendmail.php) and I would like to make it simpler for user and prevent any page refresh or transfer and output all results on the same page instead. I know that this is doable and in fact pretty simple for those who is familiar with Jquery and ajax. Also my form is using one method of validation to prevent spam, which is captcha image through another file recaptchalib.php, that I would also like to include in the new form.

I hope that all above is easy to understand.