Tomboy Adventure

Going home - Finish the main story for Tomboy Adventure.

You finish the story of the game by interacting with Natsumi, until you can finally confess your feelings. Interactions gain relationship points and change your relationship status, which also unlocks different interactions like headpats or princess carry.

Headspace - Ignore personal boundaries, face the consequences.

When you unlock headpats, keep patting her head until she punches you.

Like a princess - Carry your princess around.

When you unlock the princess carry, just pick her up.

Revenge - Get pushed into water.

When you unlock the princess carry, pick her up and drop her into the water. I found the pier at the start of the game to be a good spot to do this. Now she will be up for revenge. Stand at the edge of the pier and she will push you into the water.

Treasure hunt - Find the hidden treasure.

In the hut near the start is a treasure map. It leads you to another hut where the next treasure map is. Follow the mark and right next to the water will be a third treasure map. At the final spot will be a note. Give it to Natsumi to get the achievement.
Note: I noticed that the next treasure map only appears after the dialog for the current one is done. So make sure that Natsumi is near you when you pick up a map/the note. Otherwise the dialog might not play.