The first sentence uses the structure "go + gerund", which is common with outdoor activities for instance. Other examples will be: go swimming, go shopping, go diving, go driving.
The second sentence is not correct. You can say "I went to ride a bike" or "I went riding", as in the first sentence; although if there is an object I would expect the structure "go + infinitive (with "to") to appear more often than "go + gerund". In other words, I would say "I went to ride a bike" and "I went riding", and not so often "I went riding a bike", which in my opinion is not incorrect, but less common.
The third sentence has the same concept as the first one; it is only in the present.
In the fourth sentence "go ride" means "go and ride". That is why the infinitive is possible, because it is "Let's go" and "Let's ride". It is similar to sentences like "We should go water the plants." "He wants to go see the film." "I will go tell them."