Public transport in LA is not good, so we always hire a car. We hired a car via Alamo. For 3 days, CDW and 2nd registered driver, the bill came to about US$135. It would be cheaper without the second registered driver and this was based on the Alamo $19 per day weekend deal (exc CDW). Petrol was running at $2.87 - $3.13 per gallon and we probably used about $25 in gas for the 3 days.
From DLR, you would go north on the I5, West on the 91, North on the 110, West on the 10. Take the Fairfax Avenue exit on the 10 and stay on this till 3rd Street. Head East on 3rd Street and the Farmer's Market will be on your Left hand side.
We drove 20 minutes from LAX to Farmer's Market and then 30 minutes from Farmer's Market to Long Beach; both of which were after 9 am and before 4 pm in the day. I can't remember the last time I drove from Anaheim to Farmer's Market but I would hazard a guess that this may take you about 30 minutes to do, depending on traffic/time of day.