Surprisingly there are a handful of third party Wii games with their official servers still online. The only one I have is Goldeneye.
Connecting the Wii to Wiimmfi is pretty easy. If your Wii isn't softmodded, you have two options.
Method 1: https://www.youtube.com/watch?v=yhJOCfmyhXc
Method 2: https://www.youtube.com/watch?v=3CBSQ9Afqiw
Method 1 is a bit buggier and requires the Wii Internet Channel (which won't be available after this month because the Wii Shop Channel is closing). Method 2 is more reliable but you have to wait a minute or two for the game to start every time.
The main Wii games I play online are Mario Kart Wii and Tatsunoko VS Capcom. Unfortunately, Monster Hunter Tri hasn't been revived, and all projects attempting to do so have been abandoned. They need to get on Shu's level