As of Swift 3 / 4, this is done as follows.

To add a new element to the end of an Array.

anArray.append("This String")

To append a different Array to the end of your Array.

anArray += ["Moar", "Strings"]
anArray.append(contentsOf: ["Moar", "Strings"])

To insert a new element into your Array.

anArray.insert("This String", at: 0)

To insert the contents of a different Array into your Array.

anArray.insert(contentsOf: ["Moar", "Strings"], at: 0)