This is not a jungling build, it's for those who want to lane with Shyvanna in classic, and help butcher team fights in dominion.

This build is for a shyvanna who does well early game, since she naturally hits a little heavier, she becomes a strong laner that can harass and push back the enemy team with her aoe, and twin bite, then mid game, be a decent dps 1v1, then end game, being a tanky aoe dps who can initiate fights, and force some positioning with popping her dragon, and burning them all down at once while sustaining her health pool.

For runes I chose 9 greater mark of alicricity, for increased attack speed, so I could lengthen burnouts up time, and reduce twin bites downtime.
9 greater mark of vitality's, for better survivability end game.
9 greater seal of potency since it stacks so well with your burnout, and flame breath for increased damage output, and health vamp end game.
Then 3 greater quintessence of alicricity's, again for more attack speed, to up the time of burnout being effective for mobility and damage, and reduce the twin bites downtime.

Since you're an ap and ad class, I only went up to get ignite, and magic pen in the first tree, then went to get my xp rate up, flash cd down, mobility up, and gold up. I avoided getting the things that regenerated your mana, since Shyvanna doesn't have mana. Although, I took the 2 points in the %of health up, even though it has mana, since it was the next best thing to get. This build works best for the aoe dps build, since burnout is ap.

I start with boots of speed, and 3x's health potions. Due to it stacks well with burnout, and gives you the mobility to get the kill, or get away from being killed. Also allows you to stay in lane enough to kill enough minions to farm for your first items hastily.

I upgrade to mercury treads next. I do this because, the less CC they can put on you, the more you can keep your mobility up to get the kill, or run away when necessary again. Seeing as you're lacking vamp regen early game.

I then work for wits end next. Either getting the MD first for it if I'm vsing an AB comp lane, or if I don't need the extra survivability, I get the recurve bow since it helps you keep your burnout up very well, you hit naturally heavier early game, and it goes well with twin bite and it's cd.

After wits end I get my first vamp item, wriggle's lantern. Usually I grab the vampiric scepter right after wits end to give me more life sustainability, then upgrade, to further increase my survivability so I can burn health a bit more. It also gives you a bit of ad, which since Shyvanna uses both, so it's a noticeable difference whenever you get either.

I work for Guinsoo's Rageblade next, since it increases my haste further, which lets me use my burnout for a longer period of time, along with shortens the cd of my twin bite faster, and also gives me some ap and ad. Which is essential to my damage output.

I grab hextech revolver as soon as I can after that. To give me the spell vamp I need to increase my survivability more for team fights. Basically it gives you aoe health regen, the more opponents the better when your burnout is activated.

After you upgrade to Hextech Gunblade, you should be noticing how hard you are hitting, with how much health you are draining at the same time. Seeing as it gives you a boosted ad vamp, and a great ap vamp. Which lets you get health back with every move you make. Then with your haste increased, it makes it so the cd's are short enough for you to spam enough aoe damage that you are very hard to kill at this point.

I get trinity last, since it just rounds it off. It gives you more attack speed, ap, ad, a little bit of health, increased movement speed, chance to slow enemies, and also every time you use an ability, your next auto attack does 150% damage. Which basically makes sure you hit hard, have great surivability, great mobility, and great chasing ability, so it just rounds off the build, buffs each thing a little bit more, gives you that chance to crit, and makes you a winner for team fights.

I take burnout first because it lets me get in do some damage to everything, get some minion kills, and helps me put more pressure on the other teams champions early on, and also gives me the advantage of getting away since it stacks with my boots of speed well. I max it first since it's my main opener, and will get some serious damage out + increased mobility.

Then I take Twin bite, it's basically your big hit move, that's good all throughout the game, except not quite as effective early game before you get wits end. So I max it second since it's a great mid-late game move.

I grab a point of flame breath at level three, then max it last. It's good for debuffing and giving you that extra push you need later, but early on, before your dragonform can spread the Aoe debuff to give your team that extra advantage, it isn't as effective as the other two moves.

Obviously you take a point in your dragon form every chance you get just because that's basically your bread and butter later when it comes to burning someone down and getting the kill.

Ignite - you don't really have a finishing blow, so early game spam this when they're low, so you don't have to turret dive to get that last hit in, or if you miss your skill shot kill.

Flash - good for getting away if you decide to charge in with your dragon form over a treeline, and it ends up being a losing fight.

Sprint - used more for jungling for the gank, rather than being an aoe dps. Good for escaping team fights when you get slowed.

other spells are usable, and it's up to preference, although I use Ignite and Flash for Dom and classic, just do to the fact it works best with the moves she has.

Pro's
High mobility
high damage without many damage increasing items
great team fighter
can get out of bad situations easily

Cons
1v1 is difficult without minions around to sustain you
skill shot is easily blocked
has problems vsing tank classes

With my build, mid to late game, you are basically an aoe vamp tank. Lead in with dragon form then burn out if it's up, or just lead in with burnout, then your flamebreath, and spam q every chance you get, attempting to leave the team in front of you. This way your attacks do the most damage possible with the debuff, you have sustained health regeneration with your two aoe attacks, and you have massive damage output.
All in all, you have a semi low health pool, but your regen from attacks makes up heavily for it, making it seem like you have a lot more brawn than you should.

All in all, with this build, you want to avoid 1v1 fights end game with characters like akali, vayne, etc. Classes that are carries and can troll you down. You are made to break the other team in team fights, and due massive damage to everyone, while you keep yourself up with all the vamp you have for spell and damage.