To get to Burmecia your party must first pass through Gizamaluke's Grotto. There are a few optional things that you can do before heading to Gizamaluke's Grotto. I suggest doing these if you want to complete all of the Side quests. You can get another party member at this point if you would like, so that you can have four characters for this part. You can get this character later on.
Move north until you come across a patch of trees, and when you approach it it is marked with a '?'. This is Qu's Marsh, where you can find Quina and have him/her/whateveritis join your party. Move up the path, you can talk to Mogster here if you want to read over some information again. Move through the path in the grass behind him and follow it until you come to a pond with frogs. That thing is Quina... You have to catch a frog and give it to him/her and h/she will join your party.
Just wait until a frog jumps onto land and then run towards it and press X. If you went exploring around Alexandria Castle you have probably seen Quina already. Leave this area and head out of Qu's Marsh. You can read more about the Frog Catching mini game in the Side Quest section of this guide.
Now that you are on the world map, you can go and visit the Chocobo's Forest to gain the ability to call and ride chocobos. Chocobo riding allows you to not have to fight random battles when you are on the world map while riding one.
When you are moving across the plains you can have Quina learn some rather useful abilities by having it eat the enemies when they are close to death. Head east from Qu's Marsh to find Chocobo's Forest, cross over the bridge and go into the circle of trees. You can find some Chocobo footprints outside of Chocobo's forest, just stand on them and use a Gysahl Green if you want to ride Choco. I suggest walking on foot simply because you can gain more levels before going onto Gizamaluke's Grotto.
Move east from Chocobo's forest and move up until you see the South Gate, Bohden Arch. It's marked on the map with an '?'. Move across the log to the right and search the end for Moccha Coffee. You can drink from the spring to restore your party's HP and MP. Now move up towards the gate and to the right of it search for an ether, then leave this area.
Now head back to Qu's Marsh on the world map, do not enter it. Face towards the north and follow the roots coming out of the ground. When you reach Gizamaluke's Grotto head inside. Once you gain control again head to the right to find a wounded soldier and talk to him for a Gizamaluke Bell. Move to the top of the room and exit through the doorway, ring the bell to open the door.
Fight Black Mages that Zorn and Thorn send at you, move past the bridge and battle some more of them. When you kill them you will receive another Gizamaluke Bell. Head up the steps and open the door to your left, Move all the way to the end and talk to the wounded soldier to get another Gizamaluke Bell. When you head back pick up the Bronze Vest by searching by the corner of the path.
Now move back and this time go through the door on the lower right, not the one up the stairs. When the '!' appears over your head while you are ascending up the staircase search to find Mythril Gloves. You should equip these to Freya so she can learn some new abilities. Continue following the stairs until you see a '!' then search for a Magus Hat.
Continue along until you encounter a couple of moogles. When Mogmi asks you for the Kupo Nut agree to give it to her and her husband will free himself. Open the chest that is in the middle of the bell to get another Gizamaluke Bell. Move into the door on the right and open it using the bell. You can talk to Mogmi to save your game and use a Tent if you need healing. Prepare well because you are going to be fighting a boss soon.
Talk to Moguta and tell him that you like Kupo Nuts. Don't head up the ladder, the enemies up there are too powerful for you at this point. Attempt to leave the area and Moguta will stop you and give you the Holy Bell. Now move back into the room with the bell and go into the door on the left. Remember to equip your characters with their best equipment and abilities before heading inside. Once you are inside a battle with Gizamaluke will begin.
Gizamaluke is pretty tough, he uses a lot of damaging attacks. To start with, if Zidane has The Ogre equipped use Soul Blade on Gizamaluke, this will cause most of his physical attacks to miss. Use Freya's Jump Skill each turn that she has, this deals the most damage and she can avoid Gizamaluke's attacks in the process.
Gizamaluke holds an Ice Rod, but it is hard to steal this item from him, it is probably better to just attack with Zidane instead of stealing. The Ice Rod can teach Vivi the Blizzara ability, which is quite powerful at this point. Don't worry, the Ice Rod can be bought from a shop later on.
For Vivi use either Blizzard or Thunder. Gizamaluke may silence Vivi, if this happens just have Quina use Echo Screen, assuming that you have Quina. If you do have Quina this battle is easier, use him/her to heal your party members when it is needed. Gizamaluke will sometimes use a Water attack on the entire party, this deals quite a bit of damage and can be devastating.
Continue attacking him using the method above until he goes down, he does have quite a bit of HP for this point in the game so be patient.