First World Jachol Cave ABP Grinding Location
This is the best location in the entire game for you to farm solely ABP for leveling up your jobs. Best of all, this location is available as early as World 1 but it'll become unavailable for the entirety of World 2. If you're reading this while on World 2, I recommend my other grinding location in basement of Castle Bal.
Any time after acquiring the Purple Chocobo or the Airship you'll be able to come to Jachol Cave and grind. Before coming here though it's highly recommended that you stock up Phoenix Downs because we're going to be needing a whole lot of them for this. For anyone who doesn't know where Jachol Cave is located, use my screen shot below of the World map.
In this cave you'll find two types of enemies, Nut Eaters which are the lower level, normal looking, extremely easy to defeat squirrels. Along with these guys you'll find one of Final Fantasy 5's super enemies, SkullEater. These little devils hit extremely hard and are extremely resistant to all types of magic and physical attacks. Luckily, we don't need to defeat them to get what we want from them.
The SkullEaters usually flee the battle shortly after it starts, sometimes right away. When they flee you won't get any EXP or Gil but you will get ABP which is exactly what we want. So basically what you're going to do is run around in this cave until you encounter a SkullEater. When you do, spend the entire battle sitting there and waiting for it to run, using Raise (White Magic) or Phoenix Downs on anyone in your party that it one shots.
Defeating the SkullEaters will net you 100 EXP along with some Gold but as said before, it's not required. Also, since they're so resistant to physical and magical attacks - defeating them is unlikely. In addition to being highly resistant to magical attacks, using magic against them will get them to summon 5 more SkullEaters (none of which you'll get additional rewards for).
If you decide you want to defeat the SkullEaters you fight there's two ways to go about this. The first of which is by making someone in your party a Geomancer or a Ninja. Using the Geomancer's !Earth ability against them will usually defeat them as will using a LightningSkill, FireSkill, IceSkill or some other highly damaging item with !Throw.
While it may sound counter intuitive, you shouldn't heal up your team after each fight with a SkullEater. The regular Nut Eaters in this cave aren't strong enough to defeat you on their own and healing up after each fight will be a gigantic waste of mana considering that SkullEaters have the power to one shot you regardless.