You can only use one Wii U gamepad. So the included charging dock will suffice for that.
The question you're going to have to ask yourself first is what sort of additional controllers do you want? You have two primary options here to fill out a game in local multiplayer or as alternate controller choices for yourself.
They're the Wii U Pro Controller, supported by much of the Wii U software lineup.
And the classic Wii remote and nunchuck combination, also supported by many games.
Sadly, they're not perfect substitutions for each other. For the primary difference, if a game depends on motion and the other unique functions of the Wii remote, it probably isn't going to support the Pro Controller as well.
The two big examples here that come to mind are Nintendoland and Wii Sports Club, two of the Wii U's most popular local multiplayer offerings which both depend on the Motion Plus features that Nintendo eventually introduced to their Wii remote design. And of course, only Wii remotes can work in backwards compatibility mode with Wii software. And not all Pro Controller enabled software can make use of a Wiimote and nunchuck combination as well.
Charging Pro Controllers is a bit of a pain. It takes about 4 hours (Although you're good for 80 hours or so of use afterwards) and Nintendo highly recommends that you only charge it via USB connection to the Wii U. The crux is that the Wii U itself must be powered on to charge it. Nyko has made a well received dual charging dock for charging two Pro Controllers simultaneously from a wall outlet. Looks odd, but I've read good things.
And Nintendo fairly recently finally introduced a first party charger for Wii remotes, available from their online store. Additional batteries can be bought separately. Never had much luck with 3rd party solutions here, but this one should work well.
Lastly, if you're going to be playing Wii software here, you may want to consider a controller to plug into your Wiimote for compatible software (A fair number of Wii U games also support Wii Classic Controllers, including all Virtual Console titles outside of DS games).
The original Classic Controller and the follow-on Classic Controller Pro are out of print (And beware of counterfeits with these, Wii remotes & nunchucks, and Wii U Pro Controllers), but there are two officially licensed Classic Controller equivalents currently available that are styled after GameCube controllers.
I've had good luck with the Hori design.
PDP also makes one, but they don't have a good reputation like Hori does and it even looks of lesser quality. They also haven't reviewed as well, and lack the turbo feature of the Hori option. I do though like that the right analog stick matches the left; not many seemed thrilled with the way Nintendo originally handled that on the GCN (Which Hori copied with their design).
There are a lot of other Classic Controller equivalents out there as well, some licensed, others unlicensed (The Wii is the last console that doesn't lock out unauthorized accessories).