Here's some pertinent info for arriving in Los Angeles.
Whole Foods - You know, where all the rich white people shop. Except it's LA, so everybody shops here. It's just expensive, so I only get specialty stuff here.
Gelsons - OK, Gelsons is more expensive, but I just love it. The one in Hollywood is so clean and perfectly maintained. It's worth the extra $$.
Ralph's - One of the chain stores. Pretty standard grocery store. Some are dirty, some are wonderful, depends on the neighborhood.
Vons/Pavilions - Another chain. Both are owned by same company. Some of them have Starbucks or Panda Express inside.
Ikea - There is only one Ikea in Los Angeles, and it's in Burbank. So if you don't live near the valley, plan time for the trip!
Target - My go-to. There are only a few around town, more of them exist in the valley and out of LA. A new one is being built in Hollywood, but for the meantime, the most central location is in West Hollywood.
Walmart - You have to travel a bit for a Walmart. There aren't any directly in LA. Well, there may be a few but they might be in sketchy areas...
Bed Bath and Beyond - There are a few scattered around. I hit up the one on Sunset and Vine. Free parking for an hour.
Home Goods - Love this store. There is a great one in Glendale that is surrounded by a ton of other shopping (big mall). You can get great deals!
Pier 1 - In case you're sick of Ikea, Pier 1 isn't bad for furnishing your place.