The Time-door Talisman
You are now on a quest to enter the alternate dimension mentioned in the memory banks of the Spaceship. To do this, an item was mentioned. It is said to be in the Cave of the Dead, which is where you fought the three-headed dragon. Return to this cave whose entrance is hidden by the treasure chest in the warehouse of the Subterranean City. When you enter the cave and activate a light source (either a Lamp or the Flash spell), start heading to the left until you reach a dead-end. In the middle of this room is a wall which contains ruins that you can't read. When you find this wall, walk 7 steps south and 4 steps west. You should hear a ringing sound like something can be found at your feet. Search the ground, and you will discover the Time-door Talisman.
Without a walkthrough such as this one, you'd have to do a little interviewing in order to figure out what to do next. The citizens of the Forest City begin to talk a little more now that you've accomplished so much. One person near the middle of town explains how there is a door to another dimension on the far edges of Fairyland, but he doesn't get more specific than that.
A few other women have new pieces of information. One mentions how the City of Illusion is found in the center of another dimension. Another mentions how you have to blow a Horn (such as the one you collected in the Spaceship) in order to enter the Phantasmal City. And lastly, to tie these two together, another lady by the general store mentions how the Phantasmal City is also know as the City of Illusion. Your next step is to find this dimensional door.
The dimension gate
You may have already seen the dimension gate in Fairyland and not realized that it was a dimension gate. It is located to the west of the cemetary, although the easiest way to find it is to simply head east from the Forest City and follow the river as it winds north and east until it doubles back west for the first time. You must use the Time-door Talisman to essentially equip it. If you don't, stepping into the gate will merely transport you onto the isle to the right, where you'll have to find the invisible teleport spaces that send you to where the Money Changer was located. With the Time-door Talisman equipped, you will vanish and appear in...
A Different Dimension
You will arrive through the gate near the top along the west coast of this new land, which is a very large island. There are three monsters that roam around here, one of which is good. As was the case on Fairyland, the walking plants should not be harmed. Instead, focus on attacking the Heavy Slimes and the Wights.
There are a few features on this island, including two cave entrances. The one on the right is fake and only leads to a trap with a poison arrow. The one on the left is the exit from this dimension which returns you to the dimension gate on Fairyland. Be warned: now that you've visited this dimension, the weak monsters that you once encountered throughout Fairyland are replaced by much stronger enemies.
Your goal is to find the City of Illusion that was mentioned in the Forest City. It was explained that it can be found in the center of this land. What you'll find at the center is a small island with two gravestones on it. Walk onto this island and blow the Horn, and you'll be magically transported to the City of Illusion.