If I have a staff that shoots flame will using it increase my destuction skill?

Nope, unfortunately it won't.

It does, just very, very slowly. It is most noticeable if you have a low destruction skill.

This is completly wrong,

Staves have no effect on your skills being raised, but your skills have an effect on the staves

As the higher the skill of the stave (in this case Destruction) will reduce the cost of casting the stave and enable it to be used more times before becoming empty.

Have you tested it yourself? I have and it does. Go try using a flame staff on a character with a 15 destruction. After using it exclusively for a while the skill will go up. But is SLOW. Like you would have gained 4 levels if you used destruction.

I have in the past and no, my destruction skill has never increased, or my conjuration skill as a matter of fact. All comments and instructions about staves also support my answer not yours.

As an example http://uk.answers.yahoo.com/question/index?qid=20111122045215AAuvPWy

Staves are just there to look cool.
They aren't really much use, at least if one is taking being a mage seriously (and thus reducing casting costs as low as possible).

i think the best use for them is back up magicka launchers, and follower weapons. gave two of them to farkas and he became a lightning shooting tank

Using a Dragonpreist's staff sure raises the destruction skill, and refilling it with soul gems raises your enchantment skill while you are at it.

That only works if you are only using a few schools of magic, or less, not 5. Unless you feel like changing robes and rings every 5 seconds.

Well obviously... but then I'm not really one to try using everything at once.
Alteration and Restoration are almost useless if you don't get hit... and even so, I only ever use Restoration rarely after the occasions when I DO get hit... assuming I don't die first.
So that leaves Conjuration, Destruction and Illusion... each a valid means of defeating the enemy... but not really effective to juggle together. Best to pick one and stick with it.

Restoration is pretty good against undead, you can make them flee, and use the already dead undead to chase after said fleeing undead, its pretty funny. I just like variety. Alteration is good for paralysis as well, and water breathing is good for escaping those damned chauruses in the swamps....the master restoration spell makes me laugh , makes all undead catch on fire, run away and explode, lol.