Shyvana is a Furry champion, unlike other furry champion Shyvana only needs her furry for her ultimate(Dragon's descent) so attack speed is a really smart way for her. each basic attack grants 2 furry. Max furry is 100 so if you have alot of attack speed you can use her ultimate in almost every team fight.

I always start a match with a Cloth armor(for more resistance and for Wriggle's lantern) and 5 Hp pots, this make me very strong in early game.
Wriggle's Lantern is my first core item, this item is very helpfull for Shyvana because you get some armor, Ad and life steal.
Life steal make shylvana a really strong laner, you dont have to recall because you can just gain HP from hitting minnion(do some small jungling) and because shes a furry champion you dont need to care about mana trubs.
my next item is Trinity Force. Trinity Force gives Shyvana a huge improval, if you manage to get this item in really early game you can almost 2 shot your enemies with Trinity + Twin Bite.
afther Trinity i jump for Wit's End witch gives me alot of AS(attackspeed).
then i go for The Black Cleaver for the AS, AD and Armor pen.
my last item dipends on my ememies, i go BT if i need more dmg. and if i get nuked to hard i go BV(banshess vail) or Abyssal scepter.

Twin Bite - Twin bite is really usefull early game(also late game if you have trinity) it does alot of dmg to enemie champion's, and is really usefull to last hit minnions(for farm).

Burnout - Burnout does dmg to all ememies nearby. This ability is really usefull in team fights and to farm minnions. It also grants alot of movment speed and is really helpfull when your about to chace/run away from/dive an enemy.

Flame Breath - Flame Breath is good to lower enemies in lanes. it also gives you more dmg against the target if you hit him again afther flame breath.

Dragon's Descent - Shyvanas Ultimate. Dragon's Descent is a really powerfull ultimate, it makes all of her abilities criticall in team fights.
Twin bite - hits all enemies infront of Shyvana (exreamly good if you have trinity and life steal)
burnout - gives the normal AOE(area of efect) and Shyvana leaves a trail of fire behind her where she moves.
Flame Breath - hit and go true all enemies in front of her.

Twin Bite - really good to farming and lowering your enemies, Also good to last hit an enemie champion.

Burnout - really good when your about to attack an enemie beacuse of the movment speed and DPS.

Flame Breath - good when your laning and good to finish of an enemie out ranged from your other abilies.

Dragon's Descent - exreamly good in team fights and also really good when you dive your enemies because of the "leap"
Dragon's descent also gives you more Armor and Magic resistance and a boost on your abilities.