All of the old Wii controllers and accessories still work with a Wii U, as long as the games are prepared for them.
The Wii Remote is the standard controller as it was originally released with the Wii.
The Wii Nunchuck and Wii MotionPlus are accessories to a Wii Remote - they plug into the bottom. The Nunchuck is a secondary controller with an analog stick, and the MotionPlus is a device that contains additional hardware for detecting movement more accurately. Note that they are not independent controllers; you have to plug them into a Wii Remote.
A Wii Remote Plus is simply a Wii Remote with a built-in MotionPlus - there is no other differences, and it will work as a regular Wii Remote for any game. This is usually a better choice than the standard Wii Remote, since some games require a MotionPlus, but if you aren't going to play any of those, then it doesn't really matter, and you could always get a MotionPlus at a later date if you need one.
So, do you actually need a Wii Remote? It depends on the games you're going to play. If you're going to play local multiplayer, that almost always requires another Wii Remote per extra person (Game & Wario is a rare exception to this). If you're going to play any Wii games (including , you need one Wii Remote per person.
If you're only ever going to play single-player Wii U games, then you probably don't need one - but you may still want a Wii Remote: Although almost every Wii U game can be played exclusively with the gamepad, there is at least one exception (Wii Sports Club), and more might show up later (e.g. the next Zelda game, if they keep the sword controls from Skyward Sword).
Furthermore, many games allow the use of a Wii Remote - you might prefer that for some titles, especially if it doesn't also support the Pro controller.
If none of those reasons apply to you, then you're likely good to go with the controllers you already have - and if a game comes along that makes you change your mind, the Wii controllers are still going to be available then.
Best Answer
As @Powerlord pointed out, the GameCube Super Smash Bros. Edition Controller is compatible with the original GameCube and Wii. This is confirmed on Nintendo's website (in the same link provided, very last question)
Sidebar: Always make sure to check the box before confirming your purchase! 3rd-party companies like Hori and PDP make their own "Fight pads" as well, but they're Classic Wii controllers without the analog triggers. (They have squishy triggers, but there's no analog signal, and that means tossing your shield or grab is delayed significantly.