In the Well, give the scientist a monster that knows a lightning attack.
In the Bazaar, give the guys in the lower-right corner of the Bazaar a monster that knows a fire attack.
After beating BattleRex, go to the right of the Shrine of the Starry Night, then beat the Goopi 5 times in a row.
After beating BattleRex, there will be a Traveler's Gate in the upper-right corner of the Monster Farm.
When you get your ZapBird from the MetalMan, talk the the Metally, then he will give you a traveler's gate.
In the Library, you must of had at least 100 monsters, then talk to one of the assistants, and she will open up a room for you
After beating the Starry Night Tournament, go right and left from the Egg Blesser, then beat the Goopi 5 times in a row
After beating the Starry Night Tournament, go to the Baazar near the other hidden gate, then show the guys a monster that knows "Yell"
After beating the Starry Night Tournament, go to the room with the guy that won't let you go to his traveler's gate. Read the bookshelf with the journal in it. Then talk the little girl, she will talk to her grandfather. After showing the grandfather a goldslime, he will let you use the gate.