The media industry is racing toward an Internet-TV future at a breathtaking pace. But the swift changes, highlighted by efforts from Apple Inc., Dish Network Corp. and others, are giving consumers an array of confusing options and forcing entertainment giants to confront some sober realities. (Update:Verizon to buy AOL for $4.4 billion).

Not long ago, consumers who wanted to watch “Monday Night Football” on ESPN, “Mad Men” on AMC or “Game of Thrones” on HBO knew what they had to do: shell out for a cable package that typically...