Here's how I would do it :

1/ In the blog make all posts "protected" ( can only be seen by members )
2/ Create an "awaiting approval group" with no membership to the blog and no admin access etc
3/ Make the default group for new registrations the "awaiting approval" group
4/ After approving a member I'd change their group to one that does have membership to the blog.

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Good idea. Is there any way I can batch change all current posts from public to members only?

You could use pma to change the status. If this was just for one blog then you'd need to narrow it down by 'where post_main_cat_ID = ##' (for each cat in the blog)

update evo_posts set post_status = 'protected'

Change evo_ to match your installs table prefix

Don't forget to take a backup first, I'm blonde and we don't always get it right first time ;)

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Where do I go to make this change?

In your phpmyadmin
and you find that in your cpanel

You'd need to run that in PhpMyAdmin, which should be available through your control panel

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*edit*
lol, Damn Belgians beat us to the post again :p