Alrighty this is the first build I've ever done on here so wish me luck. I know that it's a new champ and everyone has their own play styles and, I, am not in anyway trying to get you to play Shyvana a certain way, I'm simply telling you how I build her and what I believe would be ideal.

I hope you find this informative and helpful

For runes I took critical chance marks, health seals, armor glyphs, and attack speed quints.
I take the critical chance marks because I mostly build Shyvana on crits and damage. The base crit chance isn't enough and adding to it with the items I chose and the marks should make for a good amount of crits.
The health seals and the armor glyphs are for survivability and to build her semi-tanky.
I take the attack speed quints so that, whenever her ult becomes available, it is easier to build her rage meter back up since auto attacks are the fuel to the rage and the ult is on no cooldown. So the faster you can fill up the rage meter, the better.

For masteries, I take 21 in offense, 6 in defense, and 3 in in utilities. I put a point in the exhaust mastery so she becomes all the more deadly when enemies are exhausted.
If you want to build her slightly more tanky in the beginning, I recommend taking 9 in offense and 21 in defense.

ITEMS... I have quit a bit of explaining to do on this one so I'll take the time to explain why I buy each one.
I start out with a Doran's blade. It's wonderful to have the 100 extra health and the 3% life steal in early game. If you are soloing top as Shyvana, I don't recommend you take a Doran's blade though. I would take a Doran's sheild or boots and pots alternatively when you think your lane may be tough or you are soloing top.
The next time I go back which is usually when I get somewhere around 1000 gold. That way I can get my boots of speed, my dagger (later builds into Black Cleaver), and maybe a couple pots and some wards if needed.
Next I finish my boots with Berserker's Greaves. Not only are they cheap, they give you bonus attack speed which is always wonderful with Shyvana.
I finish out my Black Cleaver the next time I go back for the added damage and attack speed. The passive on Black Cleaver reduces enemy's armor when you auto attack so you can deal even more damage to them, which is, again, just wonderful.
When I build phage, it's mostly so I can get more health. The attack damage is great too, even if it isn't that much, but I buy phage so I can build it into frozen mallet and justify building Atma's Ipaler.
Atma's not only gives you more attack damage for 2% of your max health but it also gives you a bigger chance for crits and more armor.
FINISH FROZEN MALLET. Again, for the health and the added attack damage from Atma's. The slow on it also gives you the ability to slow when chasing low health targets for the finishing blow. It's isn't always easy to because it relies on auto attacks, BUT it is possible.
Phantom Dance is for the attack speed and the crit chance.
The Sunfire Cape can be substituted by guardian angel. Sunfire Cape is more of an early game item but it does give you more health and damage near the end. If you want to take Guardian Angel, it will give you more armor and more magic resist, plus the revive if you die.

I use Burnout mostly to farm minions, therefore I take it as my first ability. It's perfect for keeping other melee champions away from your minions in fear of taking too much damage. It's perfect for zoning your enemies without going out of your zone of safety to harass.
Twin bite and Flame Breath I level almost equally. If I want to further harass an enemy, I use Flame Breath. It's like a big flaming fist right in their face and it's quite enjoyable to watch enemies retreat a little bit every time you hit them with it. Twin bite is effective on towers and is great for taking down a tower when you know the enemies are coming soon and you want to take it down quickly.
Shyvana's ultimate is very interesting. You can fling her into her dragon form in a straight line, knocking all of your enemies in that line back with you. While in dragon form, each skill turns into an AOE. Twin bite effects not only the target under attack but also the area in front of the target. Flame Breath turns into a cone of the single attack focused in front of Shyvana. When Burnout is activated, it leaves a trail of flames behind Shyvana.

When in a fight with an enemy, you want to lead in with Flame Breath, reducing the targets armor momentarily. Activate Twin Bite next then if they stay close use Burnout to inflict even more damage. If you start taking more damage then you are comfortable with, activate Burnout and leap at the enemy with you ult. Burnout will scorch the ground beneath you and also increase your movement speed. If you feel you should run back at that point then retreat and when the enemy goes for a chase, they will take damage from the flames left of Burnout. Use Flame Breath to do more damage and if you feel at that point you could take them, chase them back and use Twin Bite to damage them down.

In team fights, you are the tank. Your job is to absorb the damage and keep your team from getting picked off. Initiate with your ult. If you feel you can flash behind them and push your enemies into your team then try doing that, but it's more of a dangerous move. If failed, you just flashed into the enemy team and they will most likely beat you to death. Then your team would be left without a tank and initiator, which is never good.

If you see your team mate being chased up lane or through the jungle at low health, run towards him and ult into your enemies to keep them away. Activate Burnout while retreating to cause your enemies to have to take a different path.

All comments are welcome and so is constructive criticism but like I said, this is how I play her. It is merely a suggestion to you.