I'd like to create a directory of K2 users that can be shown on the front end of my website.

I'd like to create a registration page where users can add their name, email address, city, and fill in 3 other custom fields. These users would be members of my website and would have access to a members-only directory of other users.

New users would need to be approved by admin (me) before being able to join the website. Once approved, they would have access to the above mentioned directory (that shows all of those fields for each member) and other exclusive content.

Can I use K2 to achieve this?

Yes you can.
The custom users fields should be stored via a K2 plugin, you can use this plugin as a starting point - github.com/getk2/user-extended-fields-for-k2

You might need to tweak it to your needs. From that point everything else ACL/ User approval is pretty much standard Joomla!.

The list of the users can be generated via a Latest Items by Users menu or with a K2 Users module.

I installed K2 and the extended fields plugin. How do I edit/change the fields to the ones I need?

Thanks!

You can use this plugin as base to develop your own.
Just as any other plugin the fields are defined in the .xml file.

If you do not wish to code, you can look at the JED or the KED for a similar extension.