Do the two phrases mean what I proposed below?
Let's go grab some lunch
Meaning: go and have a quick lunch, maybe sandwiches or something similar.
Let's go pick up some lunch
Meaning: go and pick up some take-away and have it at our home or office or whatever.
I would say yes, although the first expression (grab some lunch) could also have the second meaning (that is, "grab some lunch" can also mean "order some lunch to go, and eat it someplace else").
There are other ways to express this, too, such as, "Let's go get a quick bite to eat." That could mean dine in, or carry out. Get a quick bite usually implies it will be more of a hurried than a leisurely lunch, although not always. Informal expressions like these can often be used in a more clichéd manner. I might say to a co-worker, "You wanna go grab a quick bite to eat?" and we could still be at a luncheonette more than an hour later. In that case, I wasn't really emphasizing quick, I was just using a familiar expression to propose doing lunch together.
