Doom Gaze
The trick to being able to fight Doom Gaze is to realize that it isn't a random battle in the usual sense of the word. Every time you take off in the airship Doom Gaze has a fixed location on the world map he is hiding in. To get into a fight with Doom Gaze all you need to do is find that location and get close enough to it to trigger a battle. Hence, if you're ready to fight, flying around in circles is pointless and wandering around aimlessly may not work too well either. From when you take off, tilt your ship to the right at a very slight angle, then hold L+R to go full steam ahead. You'll move across the map at a slight angle and cover every location until you run into Doom Gaze.
However, there is one time you can count on fighting Doom Gaze: the first time you leave Kefka's Tower, you will fight him, so make sure you leave the Tower at least once.
For actually fighting Doom Gaze, he absorbs most elemental spells, so I would recommend using your strongest physical attacks like Chainsaw or non-elemental attacks like Bum Rush . He's weak against Fire and Pearl however, so you can try your strongest fire spell or Pearl if you have it. And unblockable magic spells like Ultima are always a plus of course. Doom Gaze has 55555 HP total and although he usually escapes before a battle is finished he's unable to regenerate his HP. Meaning, that all you have to do to wear him down is keep fighting him over and over and do some damage each time. And since you know how to find Doom Gaze with the above method, this should be a cinch.
Doom Gaze is vulnerable to the Vanish-Doom/X-Zone combo, but if you use Vanish then X-Zone, you will not receive the Bahamut esper. There is also a chance you won't receive it when using Vanish then Doom. Save your game if you want to be on the safe side.
The Eight Dragons
For all the Dragons you can do the unworthy way of defeating them by the vanish - doom technique. May not always work on Skull Dragon but try the vanish - life technique.
Dirt Dragon
This is the one Dragon you won't encounter on any of the normal sidequests. To find it, go to the Opera House, which is in the thin middle section of the left middle continent. Go up to the balcony like you did for the original opera house scene, and flip the second switch from the right to drop down onto the stage. To actually fight the dragon, make sure you have Cherub Down equipped on every party member, which can be bought in the nearby city of Jidoor. This will give each member permanent float status which will make you immune to most of the earth based attacks and make the Dirt Dragon unable to permanently bring you to the ground either. You can also equip your party members with Gaia Gear which absorbs earth attacks. It's weak to wind and water magic, and it is also vulnerable to Poison, Sleep, and Slow. You should just use your strongest attacks and take your time and you should be fine.
Storm Dragon
Located in Mount Zozo in Cyan's sidequest. Weak against fire, absorbs bolt magic. 42000 HP total.
Red Dragon
Located in the Phoenix cave in Locke's sidequest. Weak against water and ice, vulnerable to Poison, Confusion, and Slow, absorbs fire magic.
White Dragon
Located in a top side room in the Fanatic's Tower. Equip wall rings on your entire party as usual in the tower. It absorbs pearl magic and has no elemental weaknesses. It is vulnerable to mute. Use unblockable spells like Ultima or your strongest other magic.
Blue Dragon
Located in the secret room in the ancient castle in Edgar's sidequest/Ancient Castle. It is weak against lightning, and also vulnerable to poison and slow. Use your best physical or magical attacks as usual, and you should have the offering to help you out a lot.
Ice Dragon
Located on the snowy hills of Narshe where you fought Kefka, in Mog/Umaro's sidequest. Use your strongest fire magic and it absorbs ice naturally. It is vulnerable to Poison, Silence, Berserk, Confuse, and Slow.
Gold Dragon
Fought by your second party in Kefka's Tower. Attacks with lightning, weak against water, vulnerable to poison and berzerk, by this point you should have enough powerful magic and attacks to finish it.
Skull Dragon
Fought by your third party in Kefka's Tower. An undead dragon with annoying status altering spells. Ribbons help naturally, and use your strongest fire spells or pearl.