Highest EXP/AP gains:
For simply getting as much AP and EXP as fast as possible, the Swamp area of the Northern Crater is the best because you can fight the Movers (which give very high AP) and Magic Pots (high EXP gains) there.
You'll need to give lots of Elixirs to the Magic Pots, but you can just duplicate them using the W-Item materia bug: equip W-Item, select Elixir, use it on someone or a Magic Pot, then for your second item select Elixir again, then deselect it and you'll have 1 more in your inventory. Select Elixir again, deselect it again, and you'll have another; keep repeating this as necessary.
Character Stat Boosting with Sources
If you want also wish to improve your characters by collecting sources (Power, Magic, Mind, etc.), the best area to gain levels and source items is the Sunken Gelnika.
Sources:
As a recap, sources come in several varieties, and when you use a source item on one of the characters, one of that character's stats will be increased by 1 point (depending on the type of source). Sources are very rare in the game, and there are a mere handful of enemies which morph into sources, but the Gelnika is the one area in the game where all enemies you find can be morphed into sources, and each type of source can be gotten by morphing at least one type of enemy there.
Here's the list:
Power source = +1 Strength (do more physical damage)
Guard source = +1 Vitality (take less physical damage)
Magic source = +1 Magic (do more magical damage)
*Mind source = +1 Spirit (take less magical damage)
Speed source = +1 Dexterity (get faster turns in battle)
Luck source = +1 Luck (have more luck, where luck is a factor)
*A special note about Mind sources and Spirit: since the game is bugged, the Magical Defense attribute of equipped armor is never used in the game's calculations, so Spirit is the only thing which actually protects your characters from magic attacks. Thus, Spirit is a very important stat, and the only way to increase your defense against magic attacks is to use Mind sources.
Morphing Enemies into Sources
1. Get Morph
You get it from the Temple of the Ancients. It is missable so if you didn't get it then...too bad.
2. Get Yuffie
Since Yuffie is an optional character, I guess it's possible for you to not have added her to your party, so refer to this guide to find her. Search for {OPT-1} to jump to the relevant section.
3. Go to the Sunken Gelnika
Get in the Submarine, travel to the south of Costa del Sol near the Gold Saucer. Dive and look out for a sunken airplane, while making sure to avoid the Emerald Weapon if it's lurking nearby. Steer your submarine into the sunken ship to enter it.
4. Get the Conformer
The Conformer is found on the Gelnika, which is nice and convenient, since you'll need it. You can find the Conformer hiding in room straight ahead of the room with the Save Point. The Conformer is in the chest on the lower level of the room, in the upper-left corner, hidden from view by a pipe.
5. Morph enemies with the Conformer
The method of obtaining unlimited sources on the Gelnika is to use the Morph command (gained by equipping the Morph materia) on the enemies you fight here. If you kill any of the enemies you find on the Gelnika with the Morph command, they will be transformed into a source, depending on which type of enemy you killed.
Normally, the Morph command does 1/8th the damage your normal attack would have done, which would make getting sources very difficult indeed. The trick to Morphing enemies easily is to have Yuffie in your party, and equip her with her ultimate weapon, the Conformer.
The Conformer does damage related to the level of the enemy you attack with it, bypassing many normal damage calculations, including the damage-reducing effect of Morph. Since the enemies on the Gelnika are all at very high levels, Yuffie's Conformer can easily hit them with Morph for 7000-8000 damage, making it easy for her Morph attack to be the killing blow.
Which enemies give which sources
Power sources come from Unknowns
Guard sources come from Unknown2s
Magic sources come from Unknown3s
Mind sources come from Serpents
Speed sources come from Poodlers
Luck sources come from Bad Raps
Other places to Morph enemies into sources
There are other places to get sources which may be more effective if you are looking to only increase 1 type of stat quickly. For more details, refer to this guide and search for {PER-2} which will take you to the stat maxing section.