1. how do elves learn to fight?
Slowly
Seriously: Since they have about a century to become what elves consider adults (they become what a human would call adult at about 20 - 25 years of age, but there's a phase of spiritual growth between that and the elven concept of being adult, which doesn't have a human equivalent), so they learn to use sword and bow because they try out everything.
They don't just use them to kill things, they also combine their appreciation for art with their swordplay, and do choreographed duels. Archery is useful for hunting as well as warfare, so it's double useful.
They will turn their prowess with weapons - and with other things that can be used in combat, including magic, and even the arts - to martial ends, of course, since they have enemies: Orcs, goblinoids and other "green races", many humans (nothing new here, everyone counts some humans among his enemies), and, of course, drow. Since they have time to spare, they think it's unwise not to prepare everyone for fights, since it's always possible that an enemy overcomes their combatants and starts to slaughter the non-combatant populace, and not spending 5 out of 100 years to prepare for that eventuality would be extremely unwise and short-sighted (and elves are anything but shortsighted)
2. how do elves organize an army?
Somewhat less organised than your average human, I'd say, and a lot less regimented than dwarves. It isn't necessarily a bad thing, though, since elves probably know where they belong in a fight. Their chaotic streak would balk at strict orders, anyway, and they often use smaller warbands and guerilla tactics against enemies.
There can be elven armies, of course, and I guess seeing a whole army of elves training their "kata" (for want of a better word) simultaneously (remember, they had ample time, and inclination, to study what is both useful in war and as a form of art) for the enemies to see can probably be a disheartening sight.
They will naturally settle into the positions they're best at, companies and the like will naturally form among extended family lines and tribal connections, so it will not be that much less organised than other races, and their natural streak for chaos, including change (and an aptitude for improvisation) will, of course, help.
3. how do elves go to war?
Some of this I answered above: They try small bands and surprise attacks - maybe the enemy will abandon his attack when he becomes frustrated at the unseen enemies harrying him.
If they must, they will employ large armies. The armies will have more unregular troups than normal. Most troups will carry both sword and bow, so they can change from archery to melee as the situation demands. Also, they will have rather a lot wizards there, and bards will probably do their part, too.
4. What is the elvish Art of war?(ie. strategic outlook)
They're usually smart but frail, so they make extensive use of tactics, building on their advantages. Not just guerilla warfare, but also manoeuvering during the fight, beyond a simple "CHARGE!" Elves are more likely to employ fancy tactics than your average army.
5. What are the core values of the elves
They like to blend with nature, not just (ab)use it like many other races do.
They appreciate art, and usually combine the practical with the beautiful. Their swords aren't just hurting implements, they're elaborate. Their elite aren't just very brawny warriors (not that this is ineffective - it's just not their way) but bladesingers and -dancers, who combine art with warcraft, and maybe even throw magic into the mix.
They like to take the big picture into view. They're patient, less rushed than humans (which can be a blessing and a curse, often at the same time), because they usually have the time to do so. They aim for perfection, doing a great job in a month when a passable job could be done in a week.
It is often said that elves don't just use magic, they're part of it. So unlike many other races (and/or cultures), there is virtually no reluctance toward magic. They don't mind using magic to do even mundane tasks if it's more effective than mundane means.