Shyvana is a melee champion that heavily rewards aggression. She does so by utilizing a modified Fury* resource that accrues anytime Shyvana lands an auto-attack. Once her Fury is full, she can unleash it to transform into a dragon, giving her substantial buffs, ability enhancements and some upfront damage.

You want to start with your ranged flame breath this will strip armor off of them and shock them to how much damage you do to them. Also good if you're facing a ranged opponent. Flame breath does a ton of damage so you want to level it first.

Honestly after you get to level 6-9 it doesn't matter which order you build your skills. It's mostly situational anyway. If you need to be running around ganking then get your W maxed. If you find you are clearing lanes a lot max W and E. If you are doing a lot of hand to hand max E and Q.

You then want your escape and initiate, Burnout. Getting this second will do a couple things it will allow you to be very mobile without getting boots and will keep melee champs from engaging you hand to hand because you're burning the **** out of them.

Basic combo skill order is W, E, Q. Sometimes you might want to go E, W, Q depending on if you have clear shot for your E or not.

Twin Bite
Shyvana strikes twice on her next attack. Both attacks trigger On-Hit effects and Fury of the Dragonborn effects.
Dragon Form: Twin Bite cleaves all units in front of Shyvana.
Burnout
Shyvana deals magic damage per second to nearby enemies and her Movement Speed is greatly increased for 3 seconds. Shyvana's Movement Speed bonus is reduced over time.
Dragon Form: Burnout scorches the earth, continuing to damage enemies that stand on it.

Flame Breath
Shyvana unleashes a fireball that deals magic damage and melts the target's Armor for 4 seconds.
Dragon Form: Flame Breath engulfs all units in a cone in front of Shyvana.
Dragon's Descent (Ultimate)
Active: Shyvana transforms into a dragon and takes flight to a target location. Enemies along her path take magic damage and are knocked toward her target location.
Passive: Shyvana reinforces her scales, increasing her Armor and Magic Resist. Defensive bonuses are doubled in Dragon Form.
Fury of the Dragonborn (Passive)
Shyvana's melee attacks enhance her abilities.
Twin Bite - Reduces the cooldown by 0.5 seconds.
Burnout - Extends the duration by 1 second to a maximum of 6 seconds.
Flame Breath - Deals 20% of the ability's damage to debuffed targets.
Dragon's Descent - Attacks generate 2 Fury* and Shyvana passively gains Fury* over time while in human form.

There are those few dragons in Runeterra who have mastered the intense magical energies that course through their unique anatomies, evolving into an elusive Celestial Dragon. These powerful and enigmatic creatures spend most of their time hidden away from the lands of men. However, there are those who find themselves drawn to civilization and who take on human forms to immerse themselves in the world of humans. Occasionally, one finds itself drawn to the spirit of a human and they have relations. Sometimes, in the slimmest of odds, the union will bear fruit and that offspring will be one of the few half-dragons to have graced Runeterra. Shyvana is one such creature, born to a simple Demacian farm-girl and her Celestial Dragon father. Left to be raised by her mother, her oddities were something that couldn't be hidden from her community, though she was protected by her family. Once her family had perished in disaster, she was left alone for the first time – truly alone. Shyvana quickly became an outsider and it wasn't long before the community's fear began to put her in danger. Facing mortal danger at the business end of an angry mob, she was rescued by the traveling noble, Jarvan Lightshield. The noble was a man who strangely knew exactly what she was and took her under his wing. It wasn't long before she was offered a chance to use her natural skills in the service of Demacia, as Jarvan initiated her as one of the elite guard. Following Jarvan into the League of Legends was only natural.

Shyvana is a fierce warrior, with the blood of one of the most powerful magical predators in all creation flowing through her veins. Unlike some of her fellows, she is a subdued personality – somber, cool, and collected. However, when the moment strikes, her draconian heritage manifests and there are few that can stare her in the eyes and not feel the primal urge to flee.

“By the blood of my father, I will end you!”