The best chain grocery store in Studio City is the Ralphs on Ventura and Vineland. Gelsons is on Laurel Canyon and Riverside, they are more upscale, but priced higher on pantry items.
Whole Foods is on Riverside and Coldwater Canyon a block down from the Gelsons, and all are just off the 101 Freeway.
As kumazam said, Red Carpet in Glendale is the closest. Woodland Hills Wine is about 20 min away no traffic, and also just off the 101 freeway. But for a quick bottle, not the best prices but not the worst, there is Vendome on Cahuenga and Riverside a block from the Ralphs store and also just off the 101 and 134 freeways. Vendome does have a decent cigar store also with ok prices and a nice selection of cigars. Abdul is the one to see there. All of these places are within 5 min of each other except Red Carpet and Woodland Hills Wine.
If dining in Studio City, try Zacks Italian Cafe, its on Ventura just down from the Ralphs, they even deliver. The garlic Cheese rolls are wonderful. Le Petit Bistro is on Lankershiem just past Moorpark. All in Studio City. If you want resturants, all you have to do is cruise Ventura Blvd from Coldwater Canyon, just past the Sportsmans Lodge, to Van Nuys. They are all there. Every kind of food you could want, you can find. I reccomend checking out Jerry Deli at 3am.

Hope thats a good start.