Amazon.com offers free shipping on most orders over $25 (within the USA). Home Depot has a similar policy. I'm sure that there are plenty of other sites that do the same thing, but bear in mind that the shipping is simply added to the cost of the products. It does make shopping easier though.
Depends on what you're looking for. The reason why some things require payed shipping is because it takes money to ship items from other countries (or states) to your house in a certain amount of time. Try amazon.com. Good luck on whatever you're looking for.