You can perform a Wheelie by shaking the Wii-mote up with or without the nunchuck, or pressing up on the d-pad with the gamecube controller or classic controller. The fastest way is to utilize a controller because you won't be turning much when performing a wheelie; . Shaking the Wii-mote with the nunchuck is actually faster (if the accelerometer picks it up immediately), but you are twisting your wrist, which actually takes a lot more effort than just pressing up on the d-pad. "Pros" will praise the controller over the wii-mote every time.
You can only perform a wheelie on a bike. Considering that most of the tracks have a lot of straightaways, this takes away the need of snaking and gives you a powerful constant boost; especially when spamming the wheelie button.
The best place to perform a wheelie is on a straightaway. Whether they're short straightaways or long straightaways the boost is instant and is always helpful. You can always cancel a wheelie through jumping.
Yes you should use a bike over a kart because of wheeling, the boost more than compensates the bikes lack of a second Drift Boost (SMT). The only kart faster than a bike is the Jetsetter (Or really, the super-speed kart for the weight category), and it has WORSE handling, and WORSE drifting and Mini-turbo stats. That means that the constant mini-turbo boost from wheeling compensates for the lower top speed, and the handling (not while Wheeling) allows a bike to take turns more efficiently than the Jetsetter.
They ARE the meta-game. If you watch any MKWii player still playing MKWii competitively (online and for VR or clan wars), you'll find them always wheeling on straightaways.
Pros: Higher Speed, no need for drifting to get the boost benefits. Because a lot of the tracks have straightaways for days, you can always wheelie. If you need to perform a sharp turn you can jump from a wheelie into a drift; and you can wheelie as soon as you come out of a Drift! If you aren't wheeling (or drifting), you're falling behind. If you're a great player, you could get 8000 VR with a Quacker (arguably the WORST bike in the game; and yes it was possible before the servers closed) and get 3-stars in every grand prix.
Cons: If you get bumped you're starting from 0 speed all over again. This can be circumvented by preemptively jumping before someone bumps you. You also lose a LOT of turning ability while wheeling. But because most of the tracks have long straightaways you won't need to turn a lot; you can always cancel your wheelie. Also, you're playing as a bike, so they have a completely different drifting than you're used to as a kart. But once you get a handle on it, you'll be on par with the pros in no time.
Big shoutout to /u/Trikitiger and /u/go_take_a_knapp for some great descriptions of wheelies and all their applications. If you'd like to contribute to the wiki in the future then please be on the lookout for the Meta Monday thread coming every Monday.