Walkthrough: Even after defeating the Main Story, your adventure is not yet over; even after saving Princess Peach and escaping from Moon Kingdom, there is still much you can do with Mario. To begin, the Post Game with Mario earning access to Mushroom Kingdom. Here, Mario can find Yoshi on the top of Peach's Castle. You can also find Toadette in one of the rooms in the castle, and she is willing to give you Power Moons for achieving certain tasks.
On top of this, a player can unlock The Dark Side and Darker Side just off of Moon Kingdom by searching for and collecting more Power Moons in all of the Kingdoms found throughout the Main Story. On top of this, each given Kingdom will have newly added Power Moons that can now be found.
Although there is no set objective during the Post Game, the goal is to slowly but surely find and obtain all 999 Power Moons found throughout all Kingdoms in order to receive a very special cutscene (much like how obtaining all 120 Stars back in Super Mario 64 will allow players to find Yoshi atop Peach's Castle).