Agree with the recs for Zo and Bloom Cafe in the area...
Will you have a car and willing to drive? If so, the city is your oyster, and you might want to give a little more info re: radius willing to drive, cuisine, and price caps...
You're close to my favorite cinnamon roll danish at Beverlywood Bakery on Pico.
Factor's Deli is right down the block from the Museum of Tolerance for a decent deli if you're not interested in heading over to Junior's or up to Beverly Hills for the Nate'n'Al's experience.
You're not all that far from Pico and Westwood for the Apple Pan, Shamshiri, Ugo...
okay this may be a bit hard to find but if your nose works its very easy. On the NW corner of Pico and Robertson there is a Persian Kosher place that has a shack in the alley( the alley runs the same direction as pico just 25 feet or so to the north) you can actually enter the butcher from pico as well, you will know the place because the man behind the cash register has a hookah attached to his lips at all times. this place make the most incredible kabob rice combos to go for about $6.00 with a can of soda at lunch time. Fish come with a dill rice and meat with the quasi saffron. you can always ask for 1/2 and 1/2. Prices are higher in the evening and I recommend this place for lunch only as to the zero ambiance. Now for your nose. When you at that NW coner point it to the sky look for the smokey flume, then let your nose guide you to a little kabob osasis.. Then get you grub and find a place to eat as its takeout only. You will not regret it.
You are about 2 miles from the best sushi in Los Angeles (outside of Urasawa which is $300 pp) at Sushi Zo. Go south on Robertson to National and make a right to 9824 National (on your left just before you cross under the 10 Fwy). Call ahead for a place at the sushi bar and tell them you will be getting omakase and be prepared to enjoy one of the best meals in LA.
Phone: (310) 842-3977
Castle Heights Avenue
9824 National Blvd, Los Angeles, CA 90034
A couple thoughts in the neighborhood -- NOT what I'd call MUST try, just a couple casual-fare recommends near Pico Robertson.
TUK TUK Thai is on north side of Pico approx 3 blocks west of Robertson, solid Thai cafe, moderately priced, not a big night out, just good Thai food in the neighborhood.
VERSAILLES is on south side of Pico at LaCienega, about half a mile east of Robertson. Simple, high-volume Cuban cafe, moderately priced. Several choices on the menu for solid Cuban daily fare, although many posters here will blast Versailles because the place is always chaotic and the menu is set up to move high volumes of simple dishes, not for the exquisite dining experience.
Some general orientation:
LA is not precisely a city for pedestrians.
Places indicated above by jadorelabouffe require driving, or taxi.
Buses and other types of public transportation are sorely lacking/defficient in LA.
Lots of eateries in Pico Robertson area, with lots of kosher places as well, but you won't find any high-ends within walking distance.
For more info, please indicate your price range and/or preferences.