Are shipping expenses part of my Inventory/COGS or operating expenses?

Freight In is the shipping cost that you pay for to costs of raw materials and/or finished goods to be shipped to you from your suppliers. This is part of the cost of your Inventory and thus should be categorized as an Inventory Asset.

Freight Out is the cost of shipping your finished product out to customers. This is an operating expense that is categorized to the Shipping and Postage expense account.