Both of the Trauma Center games are pretty tough but they are awesome and a lot of people don't seem to be very aware of them, so I always try and give them a bit of a push. Next year also sees the release of the 3rd game in the series - Trauma Team, which is one of my most anticipated games on any system.
Wii games - SMG, Zelda TP, Mario Kart, SSBB, Wii Sports, Wii Play, Endless Ocean, Blue World, Wii Fit, Super Monkeyball, Rogue Trooper, Shaun White, NMH, NMH2, GH: WT, Aerosmith, Zack&Wiki, Okami, RE4, Force Unleashed, Clone Wars, Scarface, PES, Tenchu, Trauma Centre NB, Madworld, Rune Factory, Wii Sports Resort, Metroid Prime Trilogy, Muramasa, COD Reflex, Obscure, Silent Hill, Rabbids Go Home, NSMB, Bully, Tatsunoko vs Capcom, Red Steel 2, Monster Hunter Tri
Well, if you like puzzles, then you should try Zack and Wiki: Quest for Barbaros' treasure. It has some truly mind-bending puzzles, Exceptionally pretty graphics, and a ton of fun overall.
De blob is also a really good choice. The graphics are relatively pretty, the game-play is fun but not too complex, and the entire thing is overall really charming and a joy to play.
Okami is one of the prettiest games I have ever played, plus it will last you a good long time. Think of a game that looks like it was drawn by Japanese calligraphists. The gameplay is a lot of fun too, though you have to draw on the screen with the Wii-mote, which could be difficult if you don't have full control of your hands.
When you say you're not into shooters, does that mean violence/blood in general? If so, then you'd better skip this paragraph. No More Heroes is a really fun, stylized, and hilarious game. It has a lot of blood and swearing in it, so it's not for the faint of heart, but it's fun and it's got a good plot. Madworld has even more violence, blood, and langauge, but it's also one of the more fun games I've played. It does require you to shake the wii remote a lot, though.
I personally had a lot of fun with sonic unleashed, though I would recommend a gamecube controller instead. it makes things a lot easier...
Those are the ones I can come up with off the top of my head. A lot of what I would mention has already been said: Harvest moon is pretty fun, as is rune factory. Zelda is a must, and the Mario games would probably interest you.
One last thing I'll recommend is Bit.Trip Beat on wiiware. It's sort of a rhythm game mixed with pong. it does require a lot of precision though, and it's very difficult. Definitely look it up before you play.
Whatever you end up getting, have fun! The wii has something for everyone.
Well, based in your preferences. I will recomend you:
"Broken Sword: Shadows of the Templars - The Director's Cut"
It has puzzes, a good story and all you wanted.