I try to always carry my groceries in cloth bags, but despite my best efforts, I still end up with plastic shopping bags from time to time. There are plenty of ways to reuse them — as trash can liners and dog-poo holders, for instance — but even then, the bags still end up in landfills, where they can take months to hundreds of years to break down.
The best option, of course, is not using plastic bags in the first place, but if you can't cut them out entirely, you should definitely recycle them. For a primer on what can be recycled and where to take it,
Though most curbside recycling services don't handle plastic bags, many cities have drop-off centers for type 2 and 4 plastics. Most grocery bags are made from type 2 HDPE, or high-density polyethylene, and many supermarkets have bins where you can deposit these and other retail bags; just be sure to remove any string or handles.
Some drop-off centers also accept type 4 bags, which include dry-cleaning films, paper towel and toilet paper wraps, and newspaper bags. The key is finding a location nearest you and knowing what plastics they accept. Plastic Bag Recycling has an extensive list of drop-off centers, and Earth 911 lets you enter your zip code to search for different plastic recyclers in your area.