I forgot something: I think it is best to begin the relic quest soon, so that when you do Ifrit and Garuda HM you already fulfil the quest and don't have to repeat them later. If you happen to obtain the Garuda weapon, fine.

Do the quest without rush while you gear up, so that when you reach Titan HM you'll be nicely geared.

I wouldn't obsess farming Garuda, for Titan HM a tank with ilvl55 weapon with good gear is enough. King Mogle gives an ilvl 75 weapon, so you may do it while trying Titan for a change, in case Titan takes long.

Once you do Titan you'll have the relic.