The above information is all very helpful. Just to add another 2 cents, or to repeat some points.

I personally went for Ifrit's weapon. GC weapon is nearly just as good, but I really wanted the "novelty" of having Ifrit's weapon. I did not have good luck though, so it took me something like 25 runs before getting it. Until then, if you don't have GC either, I would buy the weapon that is requested of you from the Relic weapon quest (Maddening something). Do NOT turn it in right away: feel free to USE it for a bit, until you get a better weapon. Even more so if you meld'd some quicktongue 3 materia to it. The next steps to the quest require you to do content that you want to be more geared for, so there's no rush to turn in the weapon.