Plastic bags are the best thing both for the enviornment and for cost.
Paper is the absolute worst thing for landfills: studies have shown that after 20 years paper is still clogging landfills! And paper that is burned adds huge amounts of CO2 into the atmosphere.
Reusable cloth bags sound like a great idea but unless you are a granny carrying one bag home, they are stupid: I carry 25-30 bags home from the grocery, should I buy 20 cloth bags, cloth bags made of oil-dependent, fert-rich cotton, bleached with acids, dyed with chem dyes? Or should I buy 20 bags made from oil-based synthetic fabrics, putting me in a worse place than using the cheap plastic bags? Bags that are eventually wear and wind up in landfils anyway?
Consider no longer accepting plastic bags. The more that you continue to use the bags, the more stores assume you want them. By bringing your own bags with you, and by refusing plastic bags, you send a message that they're no longer wanted. However, be realistic. Some plastic bags prevent worse problems, such as leaking meat juices––use your common sense about minimizing their use.
Turn the bags into bin liners. Keep plastic bags around the house to replace trash can liners in the bathrooms, office and bedrooms. Keep them anywhere you have a small trashcan. Keep some in the laundry room to throw out pocket trash found in clothes and empty the lint filter in the dryer.