Well, it doesn't sound like you'd be doing anything wrong in the hunt itself. Altough I'm guessing you might want to make a run for it (further away or to another area) every now and then to heal up, eat and sharpen your weapon. If the Giadrome is fleeing to another area, that's a definite sign that it's taking damage
and is about to go down. Just finish it off before it can get rest (YellowMegaMan's right. This doesn't quite work.). Next time, either catch it in a trap as it is running, or get in a good combo before it runs away (or something else, I'm sure there are more ways to hinder your prey). Even tough it does get health back by running away, it should be low.
However, the hunt for the Giadrome is not the easiest quest. It is among the easiest Guild Quests, true, but there are easier quests there as well. Slaying and gathering quest might be good to get you started before your first hunt. And that's just the Guild Quests. There are a few, much easier quests available from the Village Elder, the old lady by the fire. The Giadrome hunt is actually the quest that unlocks the level 2 of the Village Quests.
Going through these easier quests not only gets you more accustomed to playing the game, but also get you started on gathering all sorts of materials for better weapons and armor. Especially armor. I recommend gathering materials for the Giaprey-set. It might take some grinding and you'll need a Giadrome Hide for the plate, but rest of the set should prove to have nice defence against attacks from the tougher monsters. Just remember to pack Hot Drinks on the quests once you lose the Cold Resistance from Mafumofu Armor.
Also, I recommend going through at least few of the weapons tutorials to see which you like best and to learn the tricks associated with each one. (My personal favourite being Long Swords that, while lacking blocking ability, have high mobility, reach, damage and also speed, but only after the first swing.)