Buy a bicycle. LA is becoming more bike friendly. While it is not the best option for cross town jaunts, you can easily run errands using a bike. Stay away from tough back roads. Instead, opt for streets that have a bike lane. There are a number of maps, made especially for LA bikers, that provide route information.
Take the express bus. Some complain that taking a bus in Los Angeles requires a great deal of time and patience. However, with the Metro Express, with travel on the freeway built into the route, this is no longer true. The bus even has a special transponder that allows it to go through red lights. The Express line is potentially one of the fastest ways to get across town. Use the official Metro website, "Trip Planner", to key in exact departure and arrival times and locations.
Carpool for recurring obligations. There are several rideshare directories on the internet. Search for one in your area.
Join a CSA (Community Supported Agriculture) to eliminate travel the grocery store or farmer’s market. If you band together with a group of neighbors, the CSA will bring any requrested produce to a centralized point in your area each week.
Join a city car sharing program. With most car share programs, you only pay by the hour or day. The rental fee includes insurance, gas, a number of free miles and parking.