Characters:

King Boo Win 50cc Star Cup
Diddy Kong Complete retro 50cc Cups
Baby Daisy 1 Star in all 50cc Wii-Cups
Dry Bones Win 100cc Leaf Cup
Mii (Outfit A) Win 100cc Special Cup
Bowser Jr. 1 Star in all 100cc Retro-Cups
Daisy Win 150cc Special Cup
Dry Bowser 1 Star in all 150cc Wii-Cups
Rosalina Have a Super Mario Galaxy savefile and she will be unlocked after 50 races or get 1 star in all Mirror Mode Wii-Cups
Funky Kong Unlock 4 Expert Staff Ghost Data in Time Trials
Baby Luigi Unlock 8 Expert Staff Ghost Data in Time Trials
Birdo Play Time Trials on 16 different courses or win 250 Wi-Fi matches.
Toadette Play Time Trials on all 32 different courses or win 1000 Wi-Fi matches.
Mii (Outfit B) Unlock all 32 Expert Staff Ghost Data in Time Trials

Karts:

Aero Glider (large kart) Attain at least 1 Star in all 150cc Retro Cups
Sprinter (Medium Kart) Unlocked 24 Expert Staff Ghosts
Blue Falcon (small kart) Win Mirror Lightning Cup
Cheep Charger (small kart) Attain at least 1 Star Rank for all 50cc Retro Grand Prix Cups
Honeycoupe (large kart) Win 150cc Lightning Cup
Piranha Prowler (large kart) Win 50cc Special Cup
Royal Racer (medium kart) Win 150cc Leaf Cup
Turbo Blooper (medium kart) Win 50cc Leaf Cup
Tiny Titan (small kart) Beat 50 opponents during Wi-Fi play or unlock one expert staff ghost

Bikes:

Quacker (small bike) Win 150cc Star Cup
Bubble Bike (small bike) Win Mirror Leaf Cup
Magikruiser (small bike) Play Time Trials on 8 different courses
Zip Zip (medium bike) Win 100cc Lightning Cup
Sneakster (medium bike) Attain at least 1 Star Rank for all 100cc Wii Grand Prix Cups
Dolphin Dasher (medium bike) Win Mirror Star Cup
Shooting Star (large bike) Win 100cc Star Cup
Phantom (large bike) Win Mirror Special Cup
Spear (large bike) Unlock 12 Expert Staff Ghost Data in Time Trials