Pros
-Abilities just go well with each other
-Her Q (Twin Bite) does a tone of damage throughout all phases (Early, Mid, Late)
-Dragon's descent is a great tool for initiating ganks, good for disrupting a team's formation in team battles, great escaping and chasing ability, and it comes back up in almost no time if you keep farming.
-Has two great escaping and chasing abilities (Burnout and Dragon's Decent)
-A very fun champion to play

Cons
-Easily harassed early game if paired with another melee champ.
-Ultimate (Dragon's descent) can be a used wrongly, especially if used to push opponents away from your team when ganking.
-Aimed most of the times in teamfights
-People can tell if you have your ultimate or not because your fury bar indicates it.

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(Active): Shyvana next autoattack will strike twice in one swift movement. The second attack will deal physical damage equal to a percentage of her total attack damage. Both attacks will trigger on-hit effects and Fury of the Dragonborn effects.

Dragon Form: Twin Bite will damage all units in front of Shyvana when she uses her next autottack instead of just her target. Each unit hit will be dealt on-hit effects and grant Fury twice.

Fury of the Dragonborn: Each auto attack reduces the cooldown by 0.5 seconds. This doesn't work on structures.

Twin bite is probably her most damage ability and put a point in this second and max it out first for maximum damage. Also, this ability is quite spammable because of Shyvana's passive.

(Active): During the next 3 seconds, Shyvana will deal magic damage each second to nearby enemies and her movement speed will be greatly increased. Shyvana's movement speed bonus itself will be reduced multiplicatively by 15% for every second that passes.

Dragon Form: Shyvana now scorches the earth where she walks, leaving a trail of fire for 5 seconds that will continually deal magic damage to enemies that pass over it.
Cooldown: 12 seconds

Fury of the Dragonborn: Each auto attack extends the duration of Burnout by 1 second to a maximum of 6 extra seconds. This works on structures.

I put a point in this ability first for bonus movement speed and constant early game damage. A good ability to farm with.

(Active): Shyvana unleashes a fireball forward in a line that will hit the first enemy it hits, dealing magic damage to it and debuffing them by reducing their armor by 15% for 4 seconds.

Dragon Form:Flame Breath engulfs all units in a cone in front of Shyvana.

Fury of the Dragonborn: Each autoattack that hits debuffed targets will deal 15% of the ability's damage as extra magic damage.

I put one point in this ability at lvl 3 and leave it till all my other abilities are maxed. I find that this ability does the least amount of damage and all you need it for is because of the debuff it does.
Passive): Shyvana reinforces her scales, increasing her armor and magic resistance. These defensive bonuses are doubled while in Dragon Form.

(Active):Shyvana transforms into a dragon and dashes to a target location. Enemies along her path take magic damage and are pushed toward her target location.
This ability has no cooldown but to use it, Shyvana needs 100 Fury. Shyvana generates 1 Fury every 1.5 seconds while on normal form. While in Dragon Form, she will lose 6 fury every second and once the Fury bar is empty she will return to her normal state.

Fury of the Dragonborn: Each of her autottacks will generate 2 Fury. This allow for Shyvana to prolong the duration of Dragon Form by attacking

Best ult in the game, hands down. This is probably one of the best abilities to use in team fights. One swoop in with this ability, your enemies will be running around disorientated. This ability will crush you're opponent's formation, it will stuff up any channeling champions(Katarina, Nunu, Karthus, etc), and it will definitely change the battle to you're team's favor. Not to mention that it is the best gank initiator.

Her abilities are quite straight forward, twin bite is her main damage dealer, burnout is her damage over time ability that also gives her a massive speed boost, and her E ability is there to deal the extra damage. As explained above, her ult is very useful in team fights, if used properly, then the battle is yours to win.

There is really no way to explain on how to use her ability other than start of with your W ability (burnout) as the extra damage helps. If they try to fight you back, shoot your E ability for the debuff and quickly use your Q, this will deal tons of damage. And remember, spam your W and your Q as much as possible, they are your two main abilities in fights.