I think here we go is more commonly used to indicate the beginning of something.
By contrast, there we go is more commonly used to indicate the completion of something.
For instance, here we go can be used when going somewhere.
Here we go on another adventure to the movies!
Here we go can also be used more negatively, almost sarcastically.
"You don't want to travel for the holidays because you don't like my mother!"
"Oh, here we go! I don't dislike you mother, their guest room just smells funny."
In the above scenario, the second speaker believes a dispute is about to unfold, which is rudely indicated by their saying here we go.
For there we go, we commonly see it at the conclusion of an action. For instance:
Grandpa was very old, and had bad knees. It was difficult for him to sit on our low couch. i heard him strain with effort as he sat. "There we go" he said, with a sigh, as he finally settled onto the squishy pillows. "They don't make them like they used to."
There we go can also be used to express approval. For instance, if you're teaching a child to ride a bicycle, when they do a good job, you could say, "There we go!" In this situation, you could also use "There you go!" However i think because the activity is together, either is appropriate.
Similarly, it could express approval of more removed events. For instance, your football team scores on TV, and you exclaim, "There we go!"
There may be other uses of these phrases, but I consider these the main ways to use them, which are actually fairly distinct, I think.
You also mentioned let's go. I think that can be used mainly to indicate that it is time to go. For instance:
"Is this a good time to go to dinner?"
"Yeah, let's go!"
It can also occasionally be used as encouragement.
For instance, as a sporting event, you can express support for your team, the Cubbies, by shouting "Let's go Cubbies!"
Best Answer
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.