Basic Drupal multipart form example with Previous and Next buttons.

gistfile1.php

<?php

/**
 * Defines total number of form steps.
 */
define('TEST_FORM_TOTAL_STEPS', 3);

/**
 * Implements hook_menu().
 */
function test_menu() {
  $items = array();
  // Form page.
  $items['test/form'] = array(
    'title' => t('Test Form'),
    'page callback' => 'test_page',
    'page arguments' => array(),
    'access arguments' => array('access content'),
  );
  // Success page.
  $items['test/success'] = array(
    'title' => t('Test successful!'),
    'page callback' => 'test_success',
    'page arguments' => array(),
    'access arguments' => array('access content'),
  );
  return $items;
}

/**
 * Fetches and displays the form.
 */
function test_page() {
  return drupal_get_form('test_form');
}

/**
 * Displays simple confirmation page after finishing the whole process.
 */
function test_success() {
  return 'OK!';
}

/**
 * Generates the form.
 */
function test_form($form_state) {

  // Set current step to 1 if user just starts the process.
  if (!isset($form_state['storage']['step'])) {
    $form_state['storage']['step'] = 1;
  }

  switch ($form_state['storage']['step']) {

    case 1:

      // Custom page title.
      drupal_set_title('Step 1');

      // All the real form fields.
      $form['step1_field1'] = array(
        '#title' => 'Step 1 Field 1',
        '#type' => 'textfield',
        '#default_value' => $form_state['storage']['values'][1]['step1_field1'],
      );

      // Add submit button(s).
      $form += _test_add_submit_buttons($form_state);

      // Not important - used only for displaying values posted so far.
      $form['posted_values'] = array(
        '#type' => 'markup',
        '#value' => _test_display_posted_values($form_state),
      );
      break;

    case 2:

      // Custom page title.
      drupal_set_title('Step 2');

      // All the real form fields.
      $form['step2_field1'] = array(
        '#title' => 'Step 2 Field 1',
        '#type' => 'textfield',
        '#default_value' => $form_state['storage']['values'][2]['step2_field1'],
      );

      // Add submit button(s).
      $form += _test_add_submit_buttons($form_state);

      // Not important - used only for displaying values posted so far.
      $form['posted_values'] = array(
        '#type' => 'markup',
        '#value' => _test_display_posted_values($form_state),
      );
      break;

    case 3:

      // Custom page title.
      drupal_set_title('Step 3');

      // All the real form fields.
      $form['step3_field1'] = array(
        '#title' => 'Step 3 Field 1',
        '#type' => 'textfield',
        '#default_value' => $form_state['storage']['values'][3]['step3_field1'],
      );

      // Add submit button(s).
      $form += _test_add_submit_buttons($form_state);

      // Not important - used only for displaying values posted so far.
      $form['posted_values'] = array(
        '#type' => 'markup',
        '#value' => _test_display_posted_values($form_state),
      );
      break;
  }
  return $form;
}

/**
 * Handles submitted form.
 */
function test_form_submit($form, &$form_state) {

  // Save all posted values for $form_state['storage']
  // to make them available in all other steps.
  $form_state['storage']['values'][$form_state['storage']['step']] = $form_state['values'];

  // Deal with current step based on which submit button was clicked.
  switch ($form_state['clicked_button']['#id']) {

    case 'edit-next':
      // Do not finally submit the form yet, rebuild it instead.
      $form_state['rebuild'] = TRUE;
      // Go to next step.
      $form_state['storage']['step']++;
      break;

    case 'edit-prev':
      // Do not finally submit the form yet, rebuild it instead.
      $form_state['rebuild'] = TRUE;
      // Go to previous step.
      $form_state['storage']['step']--;
      break;

    case 'edit-finish':
      // Do something with all form values ($form_state['storage']) before unsetting it.
      // ...
      // And then unset it.
      unset($form_state['storage']);
      // This was the final step, so redirect to success page.
      $form_state['redirect'] = 'test/success';
      break;
  }
}

/**
 * Adds relevant submit buttons based on current step.
 */
function _test_add_submit_buttons($form_state) {
  $buttons = array();
  // Display 'Prev' button on all steps excluding first.
  if ($form_state['storage']['step'] > 1) {
    $buttons['prev'] = array(
      '#type' => 'submit',
      '#value' => 'Prev',
    );
  }
  // Display 'Next' button on all steps excluding last.
  if ($form_state['storage']['step'] < TEST_FORM_TOTAL_STEPS) {
    $buttons['next'] = array(
      '#type' => 'submit',
      '#value' => 'Next',
    );
  }
  // Display 'Finish' button on the last step.
  if ($form_state['storage']['step'] == TEST_FORM_TOTAL_STEPS) {
    $buttons['finish'] = array(
      '#type' => 'submit',
      '#value' => 'Finish',
    );
  }
  return $buttons;
}

/**
 * Displays values posted so far beneath the form.
 */
function _test_display_posted_values($form_state) {
  $output  = '<p>Posted values:</p>';
  $output .= '<p>Step 1 field 1: ' . $form_state['storage']['values'][1]['step1_field1'] . '</p>';
  $output .= '<p>Step 2 field 1: ' . $form_state['storage']['values'][2]['step2_field1'] . '</p>';
  $output .= '<p>Step 3 field 1: ' . $form_state['storage']['values'][3]['step3_field1'] . '</p>';
  return $output;
}

Thanks for your example.

Thanks, but there is an error in your code:
function test_form($form_state) {
should be
function test_form($form, &$form_state)

https://api.drupal.org/api/drupal/includes!form.inc/group/form_api/7