Drupal 7: how to redesign, move stuff around, for a content type page?
Not clear with your questions ,adding some pointers on some assumptions.

Custom Way

1 . If you want to override a node structure for a specific content type

-- You can copy the node.tpl.php to your theme and rename it to node--type.tpl.php. Inside node--type.tpl.php , you will be rearranging the elements.

2 . If you want to override every node,irrespective to any content type.

-- Copy node.tpl.php inside your theme and do the rearrangement of elements.

3 . If you want to override a particular node

-- Copy the node.tpl.php and rename it to node--nodeid.tpl.php and do the rearrangement of elements

Some Helpful Links:

https://drupal.org/node/1089656

http://www.youtube.com/watch?v=dfwU0WtCuPs

Remember to clear the caches after you are done with building of templates.

Using Contrib Module

You can use panels module to override node templates without creating them. panele module will do that for you.

Some Helpful Links:

http://www.youtube.com/watch?v=_VIkE2ZS5oU