School may be almost out for summer, but theater’s not going anywhere soon. The month of June is busier than ever, with highly anticipated new shows, revivals and tours. Steppenwolf premieres a new work by a rising voice in Antoinette Nwandu; the Goodman goes big with its revival of Eugene O’Neill’s large-cast comedy Ah, Wilderness!; royal children flood the Oriental Theatre’s stage in Lincoln Center Theater’s acclaimed production of The King and I; and one of the biggest little hits of the winter, Broken Nose Theatre’s At the Table, returns for a full-cast remount. Here are a baker’s dozen shows to put on your calendar this month.