How do I paste programmatically with Swift?

You should just be able to convert those to Swift:

@IBAction func copy() {
    let pb: UIPasteboard = UIPasteboard.generalPasteboard();
    pb.string = textView.text // Or another source of text
}

@IBAction func paste() {
    let pb: UIPasteboard = UIPasteboard.generalPasteboard();
    textView.text /*(Or somewhere to put the text)*/ = pb.string
}

You can implement copy and paste method in Swift like:

// Function receives the text as argument for copying
func copyText(textToCopy : NSString)
{
    let pasteBoard    = UIPasteboard.generalPasteboard();
    pasteBoard.string = textToCopy; // Set your text here
}

// Function returns the copied string
func pasteText() -> NSString
{
    let pasteBoard    = UIPasteboard.generalPasteboard();
    println("Copied Text : \(pasteBoard.string)"); // It prints the copied text
    return pasteBoard.string!;
}