[susan53]

First of all, as your title implies, "Let's" is irrelevant here. The question is about what constructions can follow the verb GO. The answer would be the same whether the sentences started Let's go.. / I want to go... / Do you feel like going... / It's ages since we last went... - or any other possible opening.

So, as a brief answer - after "go" any of these constructions are possible. The interesting thing is their relative frequency and the contexts in which they would be used.

1. Let's go V+ ing : Here the -ing form is a gerund and the name of a leisure activity, so really a noun. There are lots like this - swimming, riding, walking , skiing, shopping, fishing etc etc etc and all are used with the verb GO - so eg I usually go shopping on Saturdays; I want to go skiing this winter; It's ages since we last went swimming; Shall we go skating this weekend? Here GO is a strange blend of a lexical verb with the literal meaning of go somewhere and a delexicalised verb which just means "perform the action indicated by the noun (compare do the shopping, have a meeting)When might this be said? Probably not immediately before doing the activity but earlier and in another place - eg at home. Notice too that the construction is limited to sports and leisure activities which are named using the - ing form. It wouldn't be possible to say eg * Let's go seeing Ann this weekend.

2. Let's go to... : This would be normal with most verbs where GO has a literal meaning - Let's go to see Ann; It's ages since we went to help Mum with her garden; Shall we go to get some petrol? It would be much less likely with the Ving activities I've discussed in point 1 above, and the only possible example I can think of would be something like I'm fed up with the swimming pool. Let's go to swim in the lake this weekend.

3a. Let's go and swim ... : This is an alternative to Let's go to.. So it could be used in any of the examples I've given above: Let's go and see Ann; It's ages since we went and helped Mum with her garden; Shall we go and get some petrol?. But here I could more easily imagine it being used with the Ving activities. For example, if you were on the beach and bored, you might say. It's too hot just to sit here - let's go and swim. Notice the difference in context from point 1 though - you're not planning in advance but are "on the spot" and want to do it immediately. In this context I could also imagine someone saying...

3b. It's too hot just to sit here - let's go swim and similarly Shall we go get some petrol? This would be a fairly informal use though, and obviously can't be used in a past context : *It's ages since we went swim.

Even in an "on the spot" context though, though I think it would be rare and you'd be much more likely to say...

4. Let's go for a swim Quite a lot of leisure activities activities have this option: It's ages since we went for a bike ride; Do you feel like going for a run? Shall we go for a walk? Do you feel like going for a game of tennis? and it can also be used with some other nouns, especially some connected with food and drink: Let's go for a pizza; It's ages since we went for a Chinese meal; Do you feel like going for an ice-cream?. Other examples might be in a therapeutic context : I'm going for a blood test; I'm going for a massage.
It can't be used though if the only way of expressing the activity is a Ving noun - so not, *Let's go for shopping.

So - all the constructions are possible, but they would be used with different verbs and nouns and in different contexts.
Hope that helps.