First you choose the order for the card types. Then each player in turn order (starting with the active player) chooses one permanent they control of the first card type. Then those permanents are sacrificed at the same time. Then each player in turn order chooses two permanents they control of the second card type. Then those permanents are sacrificed at the same time. Finally, each player in turn order chooses three permanents of the third card type, then those permanents are sacrificed at the same time.

Permanents sacrificed for one card type won’t be on the battlefield to be chosen for subsequent card types.

If any abilities trigger while Rite of Ruin is resolving, those abilities will wait until Rite of Ruin finishes resolving before going on the stack. Starting with the active player, each player puts their abilities on the stack in any order.