I'd suggest_
  • If you don't have your job yet, get Raging Strikes (BRD. LVL 4) and maybe Internal Release (PLG, LVL. 12), as a WAR you can get Invigorate (LNC, LVL. 22) or Flash (GLD, LVL.8) to help managing your TP if you over-do it with the Overpowers.
  • Meld your gear! The stats on low level gear are extremly low and strength materia I or II are very, very cheap. If you have a ring with +1 VIT you might meld a VIT I (or vice versa, since there are only STR acc <lvl. 25 I think?). It doubles your AP from those accesoires.
    Melding ACC also helps beating up higher hunting log mobs, that's what I do on the left side usually to be able to do leves/fates/mobs more than 5 level above my leveling class.
  • Keep at least your weapon up to date. Since Off-Stats and mainstats only rise by 1-2 per gear upgrade anyway and Weapon Damage usually scales with 8-9 mainstat, always make sure to have the best weapon possible. An weapon upgrade sometimes gets you +1-3 weapon damage, which equals a full upgrade of all your other gear.
  • If you really have to fight with a DPS to get hate back, just don't do it. Don't waste 2-3 hate combos on a mob you won't get anyway. He will finish the mob off, while you can secure the other ones.