Shyvana, my my my what a champion.
First off, let me just say this. Do NOT just buy attack speed items. You will die instantly and you will make your team fail in every team battle. I've seen numerous builds saying to do this and all i can do is face-palm.

Instead... Her strength lays in blasting the **** out of the entire enemy team, whilst being stupidly bulky ( literally more than most tanks are ) to laugh at any enemy champion that tries to take her down.

Important things to note down are:
Her ulti has no cooldown, but you must have 100 fury to activate it.
The ulti takes 6 fury from you per second.
In human form you get 1 fury every second.
When you attack in any form, you generate 2 fury, thus prolonging the duration of your ulti if you attack.

W can be used to farm like a boss.

The rune page on this champion really is up to preference.
For example, you can take mass crit chance and get about 20% to crit at level 1. As you can imagine that will increase the damage you can deal a LOT.
Mass amounts of attack speed help build her fury up quicker and also add a lot of DPS. You get about +50% at level 1 with these runes.
Armour penetration runes would be useful to some extent, especially with the debuff from her E. But in my opinion you shouldnt be bothered much about enemies getting armour penetration as a lot of her damage comes from magic too.

A combination of these could prove to be useful, but if you do combo them i'd leave the crit chance at a minimum of 10%.

Don't buy cooldown reduction. If you do i will slap you.

Pretty simple. I don't want to see any objections to this rune page.
8 in offense lands the 4 levels of Alacrity. As you should know, attack speed is useful.
"22 points in defence?" I hear you say. Well. Yeah. Extra magic resist. Extra armour. Extra health. Another +4% attack speed on top of Alacrity. And of course 4% less damage taken from anything. What more could you want on a champion like Shyvana?

I put just a general idea down here. But what you realy should focus on is this:
Max burnout first. The speed is amazing for escaping and catching up to fleeing champions. The damage stacks up to a lot and is very good for farming.
Max fire breath next. This ability rocks. The debuff is fairly good and in dragon form can decimate entire minion waves. In team battles a well placed shot of this can deal so much damage and de-buff all the enemy champions. Send the tryndamere in!
Twin Bite should really only be leveled when you cant level anything else. The ability is very good yes, but leveling it only increases the damage it deals very marginally. And this damage increase only really becoems noticeable at higher levels. This is mainly because the damage percent starts so high anyway.
Ulti: Level her ulti whenever you get the chance to or even when you feel like you want to. Sometimes i take burnout or flame breath instead of my ulti lvl 2 upgrade. This is because upgrading her ultimate only slightly increases the damage you do from transforming and marginally increases her defences.

Passive: Her passive basically says: attacks on your enemies = cookies. In other words, dont be a sissy. If you stay behind your teammates whilst playing this character, i urge you to stop playing as her. Get in there, and take some heads!

Since this champion is so fast anyway (just look at her base speed at lvl 1 compared to all the other champions) it kinda limits the usefulness of most summoner spells.

Lets think for a second.
W is better than Ghost. If you run ghost on Shyvana. You will be struck multiple times with a stick. You have been warned. ;)

W also reduces the usefulness of Flash.
Since everyone bums flash, not choosing it may confuse you. Well. Heres my logic:
A) If your a good player, you will avoid situations in which you need to use flash to survive.
B) With the W spell, you can outrun practicaly any other champion in the game anyway.
C) With this build, you can run behind a turret, soak up all the damage from chasing enemies and still be peachy when your safe.

Rally - go away.
Heal.. Useful at the beginning, but its usefulness goes down once you've started leveling more. Take it if you like it/your jungling.
Smite - Take it if your jungling.

Ignite: Very useful early in the game. And still extremely useful later when enemy players may get life steal/spell vamp. Use it on them before murdering them and the kill will be yours.

Exhaust: Not even useful on this champion. If your worried about slowing them, thats what frozen mallet is for. If your worried about the enemies attack. Your Shyvana, look at your armour.

Cleanse: So underrated. Removes all debuffs and decreases new ones. Very. Nice. Indeed. This stops your opponents doing so many things to you. Think of all the abilities of champions you hate getting hit by. Fiddles fear, morgannas dark binding and imagine how much weaker they'd be without those abilities. Thats what cleanse can do for you. It removes **** like ignite and exhaust too. ^

Ignore the summoner spells i havent mentioned.

Pros:
Your a dragon.
You kill everything.
You dont die.

Cons:
Ranged champions that can kite pretty well will annoy you, a lot.

The items here make Shyvana both extremely bulky and extremely powerful. Her Q and W get boosted by her bonus attack DAMAGE. This is another reason you shouldn't bother too much with attack speed items.

Frozen mallet - slows opponents. Makes you tanky as hell. Getting it early will stop you dying easily and will help get kills by stopped the enemy getting behind their turret.

Wits end - increases your DPS ( with the attack speed runes and masteries this is very funny to see early in the game ). It also increases your magic resistance even further. Very useful, a must by.

Mercury's treads - Extremely good. Tenacity and magic resist is an amazing combo for one item. Choose these boots every time. Do not be tempted to buy berserkers boots like a nubcake. The +25% is worth nothing.

Infinity Edge - When you get this you should be criting about 600's. The crit increases your damage stupidly. Always get this item as it also powers up your abilities.

Atmas and Bloodrazor. These last two items are very situational. If your enemy have a lot of physical damage, you should definately utilise the armour gained from these two items. Atmas will give you about 60 AD with this build - the crit chance is good too. Bloodrazor is usually a staple for this character in my eyes. The 4% damage on hit is... stupidly good. Take into account that twin bite applies the effect twice and thats 8% + double your attack damage straight away.

Sunfire cape should also be considered. This will make you even more bulkier and will also increase the static damage you can do with your W. Using sunfire cape and W at the same time will allow you to farm entire minions waves in a few seconds.

Alternatively you can swap atmas for the Bloodthirster. Life steal's useful on every champion and Shyvanna is not much different. The +100 atk damage at full stacks is extremely good too!
On the other hand, if your having trouble with the enemy amassing vast amounts of armour then something like black cleaver could be considered.