Let users create content by using the default (non-admin) theme sorry to ask a more conceptual question but I am really stuck.

I have a Drupal 7 site where users will sign in and then create a certain kind of content. I have the content set up ready to go.

Do I just let them have access to the /node/add/[content-type] page? If so, how can I theme this page so it has the headers / menus / footers of the rest of the website (and not the standard Drupal 'admin' view)?

I am able to add a custom node-form.tpl.php file but when I output the form using echo drupal_render_children($form); it still looks like the Drupal admin backend, and not a form that is consistent with the style of the rest of the website. I want it to look more like, say, a webform. Perhaps there a way to insert a node/add form into a panels page for example? 

I feel like there is a "Drupal way" of doing it, it must be a fairly common use case but it seems my Google fu is failing me. Any help or advice appreciated. 

TLDR; I want a node/add form to look like a standard website webform and not part of the Drupal admin/backend.

PS: I am using a custom theme and I am comfortable with custom modules.
I might misunderstand you, or you did not yet try this. There is a permission View the administration theme, which normal users should not have. 

This means that if you login as a less privileged user that has permission to [Content type]: Create new content, the add content page will look like any other front-end page already: