Once you pass and arrive at the other world, you'll see the same cutscene from the very beginning of the game. Drippy mentions that you're going to need a new wand, saying that going to Ding Dong Dell will likely turn one up. However, before you can set off, an enemy shows up and you're thrown into your first battle. Just use your basic attack to defeat the monster, then start heading your way towards Ding Dong Dell, which is the castle to your northeast.
At the gate to the city, you realize that you can't get inside because one of the guards has "brokenheartedness." Drippy mentions that you need to fix him, but you can't yet until you visit an old man in some woods to the north. You're about to be on your way to the first dungeon, so it might be wise to fight some battles on the world map and level up some. Once you're ready, enter Deep Dark Wood.