Ok, i just started playing oblivion maybe a week or so ago, and was wondering about the random oblivion gates that are around. Is there any reason to close these gates that aren't a part of the main quest?

Secondly, I'm looking for the Dragon's claw rock.... I haven't the slightest clue where it is, and i've been looking for awhile.. Can anyone give me a clue where it might be hiding

there are always some good potions and armor/weapons in the gates as well as a stone at the end which can be used to enchant an item.

As for dragon claw rock, the green arrow on your compass should lead you to it. you can also open up your active quest menu and then press "A" for your map. It will show you the spot. The rock itself is a boulder in the middle of a clearing i believe somewhere in the mountains.