The cheapest way to ship an item is decided by two things the carrier can see and you control: how much the packed parcel weighs, and how big the box is. I'd Ship That prints discounted USPS and UPS labels, and these 45 guides work through both variables for one item type at a time.
Start with the guide that matches what you are sending. Each one names the service to buy, the packaging that keeps the parcel under the next price break, and the failure mode that costs the most for that item, whether that is crushing, creasing, breakage or a dimensional-weight charge.
If you already know what the packed box weighs, go to the cost-by-weight guides instead. If you are choosing between two carriers rather than two boxes, go to the carrier comparisons.