To get into town a taxi ($33 for one or two people, $14 per person for three or more) is quickest; that's the flat fee from the airport to any spot in the French Quarter or Central Business District. Limo service is also available for rates starting at $35, and the airport shuttle is $20.
A cheap way to get to town is the Jefferson Transit Airport Express route E2-Airport, which is only $2. On weekdays, the bus runs straight down Airline Highway (US 61) to Tulane and Loyola Ave. in the New Orleans Central Business District; the trip takes 45 minutes. (From this intersection, simply walk toward the river, deeper into the central business district, and take a left, crossing Canal Street and into the French Quarter. From Canal Street one can also take the streetcar Uptown or into Midcity.
The Airport bus stop is on the second level of the airport, outside door #7 near the Delta counter on the west end of the terminal, in the median (look for the sign and bench); the stop is a fair walk from the east end baggage pickup, and you'll probably have to ask at an information desk to find it.
Many major hotels have shuttle buses from the airport. Even if you're not staying at one of those hotels, the shuttles can often be a value for those getting in to town if their destination is near one of the hotels.