What structure should be used to store data with multiple category identifiers?
If a particular place can belong to multiple categories, you need to implement many-to-many relationship between places and categories. Usually it's done with link table:

Category(category_id(PK), category_name);
Place(place_id (PK),place_name,phone,email);
Place_Category(place_id (FK to place),category_id(FK to category), PK(place_id,category_id));


You may also want to mark a particular category as a 'primary' category for a place. There are two approaches , 1st is to have primary_category_id in Place, 2nd is to add a field to Place_Category , say relation_type ...