Mario Kart has and always been about unlockable items so here is your guide to help you out.

Unlike the other installments of Mario Kart characters are not unlocked by finishing a specific cup on a specific CC class like Daisy on the Wii you have to win the 150cc Special Cup but instead all of them are unlocked at random as you win the gold trophy for each cup. You cannot though unlock more than one character per cup.
Unlockable Characters
Rosalina
Roy
Lakitu
Metal Mario
Lemmy
Larry
Toadette
Wendy
Ludwig
Morton
Pink Gold Peach
Baby Rosalina
Iggy
Mii

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Like in previous games as you place in the top 3 of the starting cup will unlock the next one in that set of cup races
Nitro Cups – Mushroom Cup, Flower Cup, Star Cup, Special Cup
Retro Cups – Shell Cup, Banana Cup, Leaf Cup, Lightning Cup

Courses
Flower Cup
Mario Circuit
Toad harbor
Twisted Mansion
Shy Guy Falls
Star Cup
Sunshine Airport
Dolphin Shoals
Electrodrome
Mount Wario
Special Cup
Cloudtop Cruise
Bone-Dry Dunes
Bower’s Castle
Rainbow Road
Banana Cup
Dry Dry Desert (GCN)
Donut Plains 3 (SNES)
Royal Raceway (N64)
DK Jungle (3DS)
Leaf Cup
Wario Stadium (DS)
Sherbet Land (GCN)
Music Park (3DS)
Yoshi Valley (N64)
Lightning Cup
Tick-Tock Clock (DS)
Piranha Plant Slide (3DS)
Grumble Volcano (Wii)
Rainbow Road (N64)

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Cart Parts are unlocked by picking up coins from any race other than the three golden parts. Each race you can get a total of 10 coins and parts unlocked at every 50 coins. At 1000 coins collected the amount of coins needed to unlock parts goes up to 100 coins each up to 2800 to unlock all 38 parts.

Special Parts
Gold Standard Kart – Earn a minimum of one star in all cups in all engine classes
Gold Tires – Beat all Staff Ghosts on every race or collect 15,000 coins
Gold Glider – Collect 10,000 coins