LadyKatana

Best answer as chosen by Asker / Ratings
Well, there are several things that you can do. If you explore Vinheim (the area where Mithos' castle is located), you can solve the puzzles there and acquire the Sacred Stone, which will allow you access to a secret dungeon in Sybak. Also, here's the information to get everyone's best weapons:

Lloyd: Defeat the Sword Dancer all three times to acquire the Yata Mirror, the Yasakani Jewel and Lloyd's best weapon, the Kusanagi Blade.

Colette: Defeat the Meltokio Coliseum in Advanced Single Mode with Colette to aquire her Super Girl title and her best weapon, the Angel Halo.

Genis: Defeat the Red Light, Green Light! Game on the Grand Tethe'alla Bridge to acquire Genis' Strategist title and his best weapon, the One World.

Raine: Defeat the Meltokio Coliseum in Advanced Single Mode with Raine to acquire her Gladiator Queen title and her best weapon, the Crystal Rod.

Kratos/Zelos: Defeat the exhibition match after the Advanced Single Mode battles in the Meltokio Coliseum with Lloyd or Zelos to acquire their best weapon, the Last Fencer. (This can only be done after Zelos betrays the party and you must have seen the previous two Seles scenes at the Coliseum beforehand.) If this is his first time through, Zelos will also acquire his Grand Champion title.

Sheena: Defeat the Meltokio Coliseum in Advanced Single Mode with Sheena to acquire her Rose of Battle title and her best weapon, the Divine Judgment.

Regal: To acquire Regal's best weapon, he must have learned all his techniques or be over Lv. 80. Go to the jails under Castle Tethe'alla and speak with the man there. He will tell Regal that his master, Levin, was also released from prison. Regal will ask Lloyd to take him to see Levin at the Hot Springs. When you have the option, choose "Of course." Fly to the Hot Springs and talk to Levin. After the scene, talk to him again and he'll give Regal his best weapon, the Dynast.

Presea: After you open the door in Vinheim and view the scene before you're able to warp to Mithos' Throne Room, leave Vinheim and fly to the Ozette Ruins. Ther will be a man there. Talk to him and watch the scene. Afterwards, talk to him again and he'll give Presea her father's ax to her, which is her best weapon, the Gaia Cleaver.

As a side note, most of these can't be received until after you get the Derris Emblem in Welgaia. Hope that helps you.