Diddy Kong (playable character)
Clear Chestnut Forest (World 2) in Challenge Road, then talk to him in the plaza.
Donkey Kong (playable character)
Clear 3 different River Survival courses, then talk to him in the plaza.
Gold Oar
In order to unlock the Gold Oar in Super Mario Party, you have to complete the 5 different routes in the River Survival Mode. After you are done completing it, you have to talk to Birdo in the plaza and she will ask you a question, which you have to choose Gold Oar instead of the Gold Bar.
Pom Pom (playable character)
Clear Salty Sea (World 5) in Challenge Road, then talk to her in the plaza.
Challenge Road (mode)...