The object of each mine is to reach the exit and get access to a deeper level. You can easily recognize the exit - it's a big wooden building at the very end of each stage. Bring some goblins to clear the path and move to the next mine!
 

Be careful, though. There's no way to return. All unclaimed resources would be lost forever. Moreover, you'll lose all the goblins and gold earned on this stage, starting a new mine from scratch. But your elixirs, gems, cards, and unlocked research bonuses remain with you! They will help you in clearing the following levels!
 
Moving from Mine to Mine is our way of doing Prestige. Each mine is a new puzzle, so you start from the beginning and find the right approach.
 
Starting from the second mine you need to activate all the shafts to enter the next level.