If your segue exists in the storyboard with a segue identifier between your two views, you can just call it programmatically using:
performSegue(withIdentifier: "mySegueID", sender: nil)
For older versions:
performSegueWithIdentifier("mySegueID", sender: nil)
You could also do:
presentViewController(nextViewController, animated: true, completion: nil)
Or if you are in a Navigation controller:
self.navigationController?.pushViewController(nextViewController, animated: true)