Some of these characters might be considered spoiler, so we've placed them after the jump.
Princess Peach: Your princess isn't in another castle, she's right there in the game. Her unlock is the reward for having finished the World Tour mode. In World Tour, the princess has been kidnapped yet again, so rescuing her from Bowser will unlock her from them on.
Toad: To unlock Toad, you simply have to link your MyNintendo account to Super Mario Run. You'll be prompted to do this early, so just sign in with your MyNintendo login information and you'll have Toad right away. He's slightly faster than Mario.
Luigi: Mario's thinner, more cowardly brother is available by trying one of the extra game modes. Head into the Mushroom Kingdom and purchase a special building called "Luigi's House." With a place to call home, Luigi will become available to use across all modes. You can find Luigi's House in the Mushroom Kingdom Shop, but it will only become available after you've collected 150 Green Toads and 150 Purple Toads in Toad Rally.
Yoshi: Like Luigi, you gain access to Yoshi by placing a building themed after the green dinosaur in your Mushroom Kingdom. After that he'll appear as a selectable character. You can find Yoshi's House in the Mushrom Kingdom Shop, after you've collected 30 Red Toads and 30 Yellow Toads in the Toad Rally.
Toadette: Similar to Luigi and Yoshi, Toadette is unlocked by obtaining a building themed after her character. You can get Toadette's House once you've collected 200 Red Toads, 200 Blue Toads, and 200 Green Toads in the Toad Rally mode.