I'll take a stab at it:
Let's go shopping, let's go swimming, let's go bowling are all specific types of physical recreation. The "go" is part of the verb phrase
go shopping and doesn't really mean "go" in the sense of movement or in the sense of "leave right now". There are lots of activity/hobby verbs like these that fit the pattern go + v-ing (See any grammar book like Azar for a list).
Let's go play seems to be a different situation. I think "go" and "play" aren't really part of the same verb phrase. They're two seperate suggestions or commands:
1. Let's go!
2. Let's play!
I think there are some missing words here that are implied. The underlying message is : Let's go (outside/over there/right now) (AND) play.
So, when we say
let's go play, we're really just leaving out the "and" between
go and
play, which is a common thing to do in informal English (at least American English). Think of all the other similar things we say:
Let's go see what Kathleen is doing. (Let's go over there to where Kathleen is so we can what she is doing.)
Could you go move the car? (Could you go outside and then move the car?)
I'm sick of you! Go fly a kite*! (
Go and fly a kite)
* Substitute your own vulgar expression, if you prefer
So, it's safe to say we often omit the
and between "go" and other verbs. We also do this after "come":
Come eat with me (Come
and eat...)
Assuming students can handle the answer (Aren't we in the elementary forum??) I would say that go +v-ing is for recreational activities. When we have two separate actions (
go and whatever else) we drop the
and in informal conversation. If students are interested and advanced, I might add: We seem to use expressions like
let's go play chess when we're talking about an immediate action, perhaps with some urgency, or when we want to imply a sense of movement, leaving one activity or place for another.
Compare:
I want to go help Aaron. (I'm going to stop what I'm doing and leave for the purpose of helping Aaron)
I want to go hiking. (Here I'm just describing a general activity I want to do.)
A related issue comes to mind; We sometimes use
and when an infinitive of purpose meaning is implied:
I went and had dinner with my friend. (I went in order to have dinner...)