Getting data from submitted form

The devel module will help a lot. It provides a dpm() function which is similar to print_r() but provides you with a nested array of data to look through in Drupal's messages area.

You're correct in that $form_state['values'] holds the user input, but since the submit handler will return back to the previous page automatically, you're better off doing something like this in your _validate() handler (or throwing a dpm() in your submit handler if you don't mind that running). This will let you debug things quickly and then submit when you're ready for it:

<?php
function my_form($form, &$form_state) {
  // Build out the $form.
}

function my_form_validate(&$form, &$form_state) {
  dpm($form_state['values');
  form_error($form, "Debugging the form.");
}

function my_form_submit(&$form, &$form_state) {
  // Perform your submit functionality here.
}