Doom Gaze
If you have not yet done so, now is the time to hunt down and defeat Doom Gaze. This will earn you the Bahamut Esper that teaches the Flare Spell. You can find more details on how to find this elusive monster on the Doom Gaze page.
Return to the Aged Man's House
Your next destination is on Dinosaur Island, the large island north of the Veldt. Bring Sabin and Gau to the Aged Man's House to trigger a cutscene about Gau's connection to the aged man who lives there. Your other party members will contribute to the cutscene as well, and Locke and Edgar in particular add their own special flair. There is no material reward, but it's a fun cutscene.
Dinosaur Island
Dinosaur Island shares most of its encounters with the Doma Area. The exception is in the one forest in the area, which you may notice kind of looks like a dinosaur when viewed from the top. This is the forest where you can fight dinosaurs. You earn more Exp on average from these Encounters than in any other space in the game. These Monsters also yield a good amount of Magic Points, but no GP. If you want to level quickly, this is the place to do it—if you can survive. These Monsters are worth a lot of Exp because they are incredibly powerful. You may want to gather all of the Treasures in Kefka's Tower before you attempt to level up here.
Tyranosaurs
Most of the time you will fight a single Tyranosaur. These are powerful Monsters with strong basic attacks. On the second round, they may use Meteor for around 1,500 damage per Character. It can use Bite for seven times its normal damage on the third turn. From there, the Tyranosaur will continue cycling through those three attacks.
Tyranosaurs are fierce and dish out a lot of damage, so you'll need to defeat them quickly. They have 12,770 HP and a weakness to Ice, so a focused attack will be effective. They lack any kind of counterattack or resistances, so go all-out.
Slightly less common is the encounter against two Tyranosaurs. To make matters worse, this will always be a pincer attack (even if you have a Back Guard equipped), so you can't run. You'll need to dish out over 25,000 damage while responding to two sets of attacks, including the possibility of two Meteor castings in a row. You'll have to be on your game, and at a high level, to win a fight like this.
Brachosaurs
The rare encounter in the forest is against a single Brachosaur, the toughest randomly encountered Monster in the game. They have a whopping 46,050 HP, and while they are vulnerable to Ice, they have high Mag.Def and very high Defense. Their basic attack is among the most powerful in the game, and their Swing special deals six times that much damage.
On any given turn they will usually attack normally, but always have another option. They can use Disaster to cause a variety of Statuses in the first round. In the second, they may use Meteor for upwards of 3,000 damage. Their special Ability for the third round is Sneeze, which will ensure someone survives the fight but could remove a vital member of your team from the battle. If you're still alive on the fourth turn, the Brachosaur might use Ultima for 6,000+ damage. Then on the fifth round they will always use Swing followed by as many as three normal attacks.
Surviving the Brachosaur's onslaught takes high levels and a bit of luck, but the fight has some amazing potential rewards. You'll actually get slightly more Exp from a pair of Tyranosaurs, but you have a chance to Steal a Ribbon from a Brachosaur and they will drop an Economizer 1/8 of the time. The Economizer is the ultimate prize here, a game-breaking relic that lets you use Spells like Ultima and Quick to your heart's content. You'll need to be able to kill Brachosaurs consistently to have a good chance to earn one, though.