Hello Mobafire!
Welcome to my guide on Shyvana, that crazy OP dragon woman!

Personally I love Shyvana, because her extremely aggressive and awesome play style really synergizes well with my love for diving and chewing through anyone I meet.

Therefore, I play Shyvana as a Tanky DPS fighter, that excels mid to late game.

P.S. This is my first guide, so any feedback will be appreciated.
Apart from trolling, then i'd just eat you.

PROS!
Amazing dammage output
Good Chasing Abilities
Amazing Late Game
Happens to turn into a dragon for lulz

CONS!
Kited somewhat easily
Focused in team fights due to nature of play style
Quite a squishy early game
No natural CC

My rune build is as follows:
9 x Greater Mark of Insight
9 x Greater Seal of Vitality
9 x Greater Glyph of Alacrity
3 x Greater Quintessence of Desolation

Now first of all, you guys must be thinking....
WHAT THE HELL MAGIC PENETRATION?
But once you think about it, Shyvana's Burnout, and her ultimate Dragon's Descent, deal magic dammage. In particular for Burnout, which I consider to be her bread and butter skill, the magic penetration is a must, as it especially gives you that small edge needed to win early game fights. In addition, the 2 core dammage dealing items within this build, Wit's End and Madred's Bloodrazer deal magic dammage, so you want as much of that dammage penetrating through.
HP/lvl Seals and Armour Pen Quints are pretty much self-explanatory, providing tankiness late game and boosting the AD portion of our dammage.
Alacrity seals should also be self explanatory, as many of Shyvana's skills synergize extremely well with attack speed.
Although if you wanted, you could get mana regen or flat mana seals. Those are actually pretty OP, but I dont use them because I dont like being cheap.

My masteries are pretty standard, 21 points in Offence and 9 in Utility.
With the utility points, I take the added experience so that I can hit lvl 6 as soon as possible, and increased buff duration, as it literally takes Shyvana a few seconds to get red buff to assist in ganks. I also take reduced death time and 1 point in HP/Mana regen, although if you want to be a cheap little *****, I suggest that you take the increased Mana, Mana Regen and Clarity.

Opening with boots and 3 Health Pots is currently the best course for Shyvana's early game, as everything is hinged on how aggressive you can be with her. The boots combined with Burnout gives you the maneuverability to terrorize whoever your laning, with the health pots available for a quick health recovery when they try their *****y harassment.

After your great early game farm with Burnout, whatever you buy next really depends upon the situation of the game. This is where my build may start to differ from others.

Generally, I would build a quick phage, as it gives a good deal of dammage and hp, and finish Mercury's Treads.

Then, depending upon whoever seems to be posing the most danger on the enemy team, I will build either Madred's Razors or Wit's End, to provide either armour or magic resist. No matter what I build first though, I will definitely build the other one second.

At this point, if I were required to take a more tank oriented role on the team, I would
complete my Frozen Mallet for the great hp and slow. If not, then I would finish my Madred's Bloodrazor for the sweet sweet attack dammage, attack speed and %hp dammage.

After the core items of Frozen Mallet, Wit's End and Madred's Bloodrazor have been built, you should be chewing through squishy opponent in a few seconds, and be able to repeatedly slice the hp away from enemy tanks.

By this stage, you should be rocking about 160+ AD. Sure this isn't as high as many other DPS champions, but remember the Wit's End and Madred's? With your Twin Bite applying on hit effects, your Twin Bites will deal an additional 82 magic dammage and 8% of their hp. Thats a lot of dammage in one hit. Your also slowing them for funzies.

The remaining 2 spots in your inventory are completely situational.
I prefer to get Stark's Fervor, as it provides great lifesteal, attack speed and armour debuff, and Atma's Impaler for the armour, and the massive dammage boost with the added 2% hp dammage.

However, if you were being destroyed by an AP loaded *****, getting a Force of Nature is viable.

Other viable items are:
Bloodthirster - Great lifesteal and attack dammage for increased DPS
Zeal -> Phantom Dancer - the added attack speed is sweet, but the movement speed is what I would get this for.
Guardian Angel - All round great tank item
Hextech Gunblade - This is actually pretty fun to play with, as it benefits from both your Armour Pen and Magic Pen runes, and the lifesteal and spellvamp mean you can gain back a whole lot of lost hp during combat.
Guinsoo's Rageblade - Havn't actually used this yet, but in theory, it would synergize well with Shyvana's skills, especially if you also got Gunblade.

ITEMS NOT TO GET
TRINITY FORCE
I see so many Shyvana's and Shyvana Guides with Trinity Force as a core item.
WASTE.
OF.
MONEY.
Sure the phage and zeal are great, but sheen is just not worth it.
Sure, sheen's passive provides some good burst, but none of Shyvana's skills scale with AP, and seriously, that extra mana would just be too OP to be fun. Building a sheen would also hamper Shyvana's already weak early game.
Because of this, my opinion is that sheen is just not worth it. The convenience of having three items in one, where one of those items, and a whole third of the cost is essentially a waste, does not outweigh the sheer uselessness of sheen as a whole for Shyvana.
Turning the Phage into a Frozen Mallet, and buying a zeal/phantom dancer would be much more effective.

For summoner spells, I take
Flash, as Shyvana benefits greatly from such a movement spell for escaping or chasing, and
Exhaust, because this is a great way to completely lock down an enemy champion, especially if you find yourself initiating with Dragon's Descent a lot. Whilst you just want to charge in there and wreak havoc, you must realise that you are not a pure tank. Sure you have high armour and magic resist, but unlike champions such as Rammus or Alistar, you do not have any active skills to aid in tanking. Don't let this deter you though, continue to charge in there! Just remember to pop exhaust on the enemy's AD carry, so that the little ***** cant do anything whilst you dismantle everyone else around him/her.

Early Game
Early game, just farm a whole lot with Burnout, and whenever you see an opportunity, harass with Burnout, Twin Bite and Flame Breath if you can land it. Generally, just one good execution of this combo will get the enemy quite low, where you can just finish them off with auto attacks, and popping exhaust or flash if needed.

Mid Game
If you didn't have a great early game, don't worry. Inform your team that you will be flash farming unoccupied lanes with Burnout whenever possible, to catch up on gold and experience, whilst they go off on ganks and pushes. Soon enough, you'll have caught up. Because that is the great thing with Shyvana. She starts off quite slow, but get her going, and she will snowball into a great mass of fiery fury before the enemy has the opportunity to **** their pants.
Ganking strategies and team fight strategies are quite similar to early game kill seeking, except for the all-important use of DRAGON SHIFTING YEAHHHHH. This ultimate is literally a game breaker. Not only is it just great for ganks and chases, allowing you to knock up and fly straights towards the little *****es, but it absolutely shines in team fights.

Why you ask me?
Because Dragon's Descent improves all your normal abilities to be more effective against multiple targets. For example, a clump of enemy champions, ripe for the picking.

How to TRANSFORM in an effective manner for team fights:
Aim you flight towards the back of the enemy team, making sure that Burnout is active to leave a flaming trails. Flying to the back brings you right up to where the squishy *****es like to hang out. Pop an exhaust if needed, use the cone of effect Flame Breath for increased dammage, and smash away at them with Twin Bites and Autoattacks. Before you know it, the cowards are either dead or running, leaving you ample time to destroy the abandoned enemy tank with your team. As you can see, this ult is devastating in winning team fights, which wins games.

Late Game
Ehhhhh pretty much the same thing, smash, own, transform blah blah blah. Except now you really need to make sure you have coordinated team fights with your allies, because losing a fight at this stage could very well lose you the match. During the mid game, being the aggressive semi-tank initiating rambo worked pretty well, but at this stage, just leave each champion to their respective roles. Let your tank walk in and take the punishment, whilst you search for the best time to destroy the enemy carrys and mages.

Another trick that can be done with Dragon's Descent, is to utilize its knockback effect to either save an ally being chased, or, what i particularly enjoy doing, to knock them into your towers.

Thanks for reading my guide on playing Shyvana, and I hope that this gives you a fresh new way to play an amazing champion.

P.S. Sorry if my guide came across as massive chunks of text, due to there not being any fancy image links and pictures of relevant stuff.
Why no pics you ask?
Because I cbf.
Yeah come at me.