Hello! I'm DankSlurpee and this is my first guide, and I apologize for the wall of text and no pretty pictures. Shyvana has become my favorite champion, and I've composed a perfect build for her. This is a build towards laning.

For Summoner Spells I take Exhaust and Flash.
Exhaust, for winning early fights. If you get fed early, you can carry and help the team far more. It also helps in sticky situations where you have to escape and Flash/your ultimate are down.
Flash, because on top of the ultimate it's nice knowing you can escape any situation with the combo. It also helps in securing kills.
Ignite instead of Flash is pretty acceptable too.
Any other spells I find pretty much do not work as well with Shyvana, and you're missing out unless you have these three.

I stick with Armor Pen. Marks and Quints, Attack Speed Seals, and CDR Per level Glyphs.
They can be exchanged for Armor/Magic resist, but this is my preference.

You're going to want to start out with Burnout and max it first. It's speed boost and AoE damage help with last hits, and a 2v2 team fight. Later on it's going to help even more with chasing down champions you've completely destroyed and be a small advantage towards team fights. Next max your Twin Bite, this really shines when you're in a fight and with it's on hit .5 CDR the attack speed comes in hand for leaning it in your favor. Lastly, max out Flame Breath, in my opinion it's a pretty weak move in regular form, and skill decently weak in dragon form, probably because it goes off of AP.
Of course though max your ultimate when ever possible, it increases your Magic Resist and the jump reall helps coral enemies,

I'd like to go a little more in depth with it too. Not only can you use it as an escape method, you can also use it to come from behind and push stray targets into your team putting tower pushing in your favor. Don't be afraid to use it to survive! Just go lane or jungle and auto-attack to get it back up again!

For items I start out with Doran's Blade for the health boost, and the small life drain to remain in the lane. I try not to go back until I have enough for Berzerker's Greaves.
I've tried the CDR boots (topped with a Brutalizer) and have found that these are far superior.
Then I start to build Trinity Force, beggining with Phage to make yourself pretty beefy, and double up with your Burnout Speed with the slow change to snag someone. Then when you go back buy your Zeal, and follow up with your Sheen, and Tri-Force. At this point, you've got a lot of health, endurance and damage, but but a Vampiric Scepter to put fights on your side, and follow up by building a Blood Thirster and get some stacks. Now from here, it's really up to opinion.
Sometimes I'll buy two Phantom Dancers and a Frozen Mallet, or instead of the Mallet another Blood Thirster. It's all about their team comp. Don't be afraid to build resistance items such as a FoN, Warmogs, or Thornmail. You may be a crazy awesome dragon but you CAN die and will likely be a first target.

Overall, just farm your best, follow the build, and you'll be carrying your team in no time.
Again sorry for the wall of text, and thank your for reading my guide (: