By Foot.
If you insist on driving, then avoid the Verrazano Br. That is the expensive bridge.
If you really are looking for the “cheapest way”, I just did the research for you:
Take the Q10 from JFK to the A train at Lefferts Blvd (free transfer). (Cost: 2.25)
Then take that to the PATH @ WTC to Newark Penn station (Cost: 1.75)
From Newark Penn, take the #62 bus to Newark (Cost: 1.50)
Total Cost = 5.50
I don’t think you can beat that.