This is what I build as Shyvana and it worth for most of the games that i played her. This build gives all that she needs: atk speed, atk damage and health, even AP.

Shyvana usually do a lot of damage through her auto attacks combined with her abilities, such as Burnout that deals damage per second.

In my opinion the best way to start with Shyvanas is buying Doran's Shield, because it will gives you armor, health and health regeneration. However this item may be sold later on, because you will need space on your inventory so you can buy Trinity Force.

Next is the boot Berserker's Greaves, which increases your movement speed and gives you atk speed. There are those people who prefers Mercury's Tread, that gives you tenacity and magic resist. It's up to you to choose which one is better.

Wit's End gives you atk speed, magic resist and deals magic damage, it's not too much damage even though it's very useful, because the damage that it deals is reduced by Magic Resist and not by Armor, so those champions whose Magic Resist isn't too high will receive a considerable damage.

Madred's Bloodrazor gives you armor, atk speed, atk damage and deals Magic Damage equal to 4% of target's maximum health. As I said before those champions whose Magic Resist is low will receive a considerable damage.

Frozen Mallet is good for survival, it gives you health, atk damage and it's passive ability is very useful, once Shyvana deals melee damage. It will make your enemy slower so you can get him very fast.

Black Cleaver gives you atk damage, atk speed and reduces target's Armor. the more you hit your target the more will be the damage that he will receive.

The last one is Trinity Force that gives you everything that Shyvana needs. Health, Atk Speed, Atk Damage, and on cast increase her base Atk damage by 150% for one attack.

Shyvana's masteries are very simple - it's just focused on damage.

Burnout is very useful to farm and deal damage. It gives movement speed so whenever you have to run away or follow someone it's just activate this ability.
Twin Bite is her most damage ability. There's nothing left to say.
Dragon's Descent gives her magic resist and armor. It also deals damage to enemies in her path.
Flame Breath deals magic damage and her passive ability makes this skill very useful, just like a bonus damage.

Flash allows you to get close to your enemies whenever they run away from you and it also allows you to run away from your enemies.

Shyvana needs atk speed and armor penetration. It's up to you to make her full atk speed or atk speed + armor penetration.
Mark of Desolation she's an AD champ, as all ad champ she needs armor penetration
Greater Glyph of Attack Speed gives you extra atk speed
Greater Seal of Attack Speed more atk speed
Greater Quintessence of Desolation more armor penetration