I suggested we go together / I suggested we went together. Which is the correct usage?

The difference between the first and second sentences are that the first uses the mandative subjunctive go and the second uses the past indicative went. You can see this by considering the sentences:

He wanted to go back to Kazakhstan, so I suggested he go with me.  
He wanted to go back to Kazakhstan, so I suggested he went with me.
He wanted to go back to Kazakhstan, so I suggested he goes with me. 

To my American ears, the third sounds just as wrong as the second. But by Googling, you can see that some U.K. speakers are now dropping the subjunctive in sentences like this. I would say that in the U.S., both the second and third stentences are definitely ungrammatical.

See this website that discusses the phenomenon.

Definitely the first.

The second would imply that you had already gone to Kazakhstan previously, and were just now suggesting that fact.