Hi and welcome to the Webmaster World!
Queries can be intimidating at first, but the SQL language (pronounced Ess-Cue-ell officially, but I prefer Squeal) is really very easy to understand.
The basic way to get info from a database is with the SELECT statement.
SELECT product_name FROM products.
this returns all the names from the products table in I/O order.
You can order your results by adding an ORDER BY clause at the end:
SELECT product_name FROM products ORDER BY product_ID.
this gives you the same results as above, but ordered by ID.
But, as you said and is proper database design, there are multiple tables. So you need to JOIN them together and get the results.
Hopefully, when the tables were designed, there is a field that is the same in both tables. In your case the mfr table is the "key" to finding the joins.
SO, to get your data, you would need to join the tables together on the same fields in both (the KEY) using a WHERE clause.
SELECT prodID, prodName, mfrName,catName FROM products, mfrs, categories WHERE products.product_id = mfrs.product_ID AND mfrs.category_ID = categories.category_ID ORDER BY prodID.
(I didn't feel like typing all your table and field names - I trust you can figure it out).
Notice that in my SELECT clause, I didn't bring in the category ID. You don't want to display that, so you don't need to bring it in your query. I did include the product ID, since we want to ORDER BY that field. Some dbs require the field to be in the SELECT in order to ORDER BY it.
Hope that gets you going. The "key" to your query is knowing which fields match across the tables and setting the WHERE clause appropriately.