Content type permission to specific users
The core's built-in support for node access control is rather course, and to control a specific role's access to view a page, it is not enough to go to admin/people/permissions to set permissions, since there is only a global setting for "View published content" that apply to all published content.

However. to give the permission to Create new content (node/add), there is no need for an extra module, since all other permissions, i.e. Create, Edit, Delete, is managed by the Drupal core and there are per-role settings in the built-in permissions interface.
First, just to make sure content is visible, navigate to admin/people/permissions and make sure you have the permission View published content checked for the authenticated user role.

Then, just below it, locate the check boxes to "Create new content" for the content types you want users beyond the administrators to be able to create, and check those for the roles you want to have this permission granted (ditto with the Edit and Delete permissions).

Also make sure new content is Published. If content is not Published, the only user that has access to it us the super admin user.

To set the default to Published for a new content type, navigate to Administration → Structure → Content types. Then click on the edit link for the type you want to configure- Locate the "Publishing option" tab. and make sure that there is a tick mark for "Published" under "Default options".

If you need fine grained access control beyond this, you need to install a node access control module.

The module I recommend for this is Content Access.  It let you control access per content type and user role.

To set the per content access settings, navigate to Administration → Structure → Content types. Then click on the edit link for the type you want to manage, and then on the Access control tab. This takes you to the control panel for role based access control settings. The screen dump below  shown how do set this for the built-in content type "Basic page":



After setting the permissions the way you want, click "Submit".

There is a nice tutorial about installing and configuring the Content Access module on YouTube. (Thanks to @shrish for bringing this to my attention.)