First thing to do is to login to the administrator area and navigate to Content > Categories > Options and then click the Tab Permissions.
Permissions tabs holds the information for each user group and any change here concerns the categories only.
From the left, you select the user group you want, and make sure that on the right the fields "Create" is Allowed, "Edit" is not allowed. With this action we ensure that the users of this user group cannot edit articles from any category but create only.
Below you will see the field "Edit Own". You must set this field to Allowed.
That's it! The users of this group can only Create new articles and edit their own articles.
With the above, you set those permissions to all categories, but you could select a category and from permissions tab make the same changes. With that way you set these setting only for a selected category or categories.