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.
Kamek's Tantalizing Tower (board)
Play all other boards at least once each.
Pom Pom (playable character)
Clear Salty Sea (World 5) in Challenge Road, then talk to her in the plaza.