All right I'm bored so I'll go a little more in depth. Your goal should be to get a 5-hit with 25% haste in gear. You should have a Magian Great Katana as well.
Atmas/Lunar Abyssites: Get these first. Atmas are key item buffs obtained from NMs in Abyssea. Atmas only work in Abyssea, and you can equip up to 3 if you have 3 Lunar Abyssites. These make a bigger difference than any piece of gear you can get. They give absurdly high bonuses like 50STR 2/tick Regain and 5% Double Attack in one Atma.
Unkai set: you get the base pieces in Abyssea. You upgrade to +1 with seals from NMs (10 for body, 8 for other pieces), and +2 with jewels from NMs (9 for body, 6 for other pieces). Each piece requires a different seal and jewel. Neck from NM in Tahronghi (low drop rate unless you trigger blue !!), back from NM in Misareaux (100% drop). There is also an Unkai earring but its not so hot. It isn't hard to get these per se, it will take you some time though.
Varangian Helm, Heafoc Mitts, Ace's Feet, Goading/Bullwhip belt all come from NMs in Abyssea. If you trigger blue weakness on the NMs which drop them, their drop rates become very high, meaning you generally get the drop in 1 or 2 kills. Twilight Head and Body both come from the Shinryu battle field (Abyssea mega boss). They seem to have sort of crappy drop rates, but you can enter the battle field as many times as you have Traverser stones.
Masamune is the top Great Katana now, then probably Hiradennotachi+2 and Keitonotachi+2 (TP Bonus). Amano falls after Masamune, but I don't know of many people bothering with relics these days. Keitonotachi+2 is easy and can be solo'd. Hiradennotachi+2 is a little harder, but can still be done by the casual player. Masamune takes a lot more investment, but it is still much easier to obtain than Amano. You get all of these GKT's via the Trial of the Magians system. You start with a crappy base weapon and do a series of quests such as defeat X mob Y times, or X NM Y times, or perform X WS on Y mob Z times, etc etc. After each quest the weapon upgrades slightly. There are still some decent weapons to be had from NMs in Abyssea, but they are generally weaker than the Magian ones.
I think that is a decent enough summary that if you look on the wiki you can get some direction.
____________________________Jacs of Leviathan
SAM, MNK, THF, NIN, SCH, DRK, BLU 90
Woodworking 100+3, Synergy 57