pmpProbeg (nice username),
Not to push it too much, but a membership at http://www.paidmembershipspro.com would give you access to some documentation that would help you.
That said, there is a "hook" in /preheaders/checkout.php that runs after a successful checkout.
There is also pmpro_after_change_membership_level if you want to run code after the membership level changes for any reason (cancelations or admin sets it).
The flow is basically preheaders/checkout.php is everything above the header on the checkout page and pages/checkout.php is everything within the checkout page. The page submits to itself to process and then redirects to /membership-confirmation/ (preheaders/confirmation.php, pages/confirmation.php).
If you want to add fields to the checkout form, you can use one of these hooks:
Or templatize the whole page by copying pages/checkout.php to your theme and renaming it membership-checkout.php (you'll have to remove the shortcode on the checkout page and add the page.php HTML around your template file).
Hope this helps you. If you need more support, please sign up for a support account.