Ok i'm just giong to point out a few things you should know:
-This build guide is NOT comprehensive. My goal in this build guide is simply to tell you who Shyvana is and how you play her properly. The builds i have include are MY personal favorite bulds for her (i have notes for the builds in the section below). You may not like these builds but it all depends on your play style. This build guide is not going to be pretty and decorated, that is stupid, it is meant for you to learn how to play shyvana. Now continue on below

These are my notes on my builds:

Build 1-Low Cost Attack Speed and survivability
Build 2-Jungle build*
Build 3-For the extremely fed shyvana*
Build 4-Offense based.
Build 5-Just for kicks attack speed

In build 2 replace Madred's Bloodrazors with Force of Nature if you are having survivability issues due to AP or Warmog's Armor if you are having just general survivability issues

Also in any build with Ionic Spark replace it with Wit's End if the other team has 3 or more AP champions

in the patch for february 7th, 2012, stark's fervor in the 4th build will be replaced with Zeke's Herald, POSSIBLY, i will have to go through the items and maybe pick out a better one because of the lack of lifesteal on the item :\

Also, Feb. 4th, i changed shyvana's jungle build. This is my favorite build and i use it the most so i recommend using my new build. My only change is that i exchanged trinity force for frozen mallet, and i will buy frozen mallet before sunfire cape. This does 2 things for me. First of all, it gives a more tanky feel. I like this because the late game attack speed (the primary reason for getting trinity force over frozen mallet) becomes unnecessary when you have frozen mallet (because of its slow). The slow is really nice mid game. Also, Trinity force is rather expesnive, and although shyvana's abilities are based on attack speed, you don't need it that much late game.

In my opinion you can play her in 4 ways:

1. Jungler
2. Laner
3. Fed
4. Laning with support champion (offensive based)

This is my personal favorite way to play Shyvana
Being my favorite way to play her i can go quite in depth with this
Build yourself in the order i have given at the top. This is the ONLY build i will tell you that you SHOULD follow. All the other builds you can change to your liking, but trust me, the jungle build i made is the best i've seen so far (keep in mind, the jungle build is build #2)

As jungle shyvana you want to start with Boots and 3 Health Potions. First you will start at wolf (that's right, no blue buff). Start with Burnout, keep hitting the lead wolf then immediately when Burnout cools down, use it again (unless you already killed all the wolves, but it's not likely). Next use 2 Health Potions. Go to the wraiths and start with the leader. Intro with Burnout then hit it until it dies. You should be all right on health by this point, about halfway i'd imagine, if not a little less than half health. Next, go to golems. Use 1 Health Potion before you engage and then engage with Smite then Burnout then Twin Bite. keep hitting until it dies and immediately recall once finished with the golems. Buy 1 Health Potion then whatever you can afford towards Wriggle's Lantern. Go attack wolves, same way as before but with twin bites as often as possible. Use your Health Potion. Go attack wraiths same way as before. You should be about half health again. go to golems. Intro with Smite then Flame Breath if you have it, then Burnout and Twin Bite as much as possible. Go back, buy more towards wriggle's then repeat the process again. On your 4th time back you can start ganking. i don't reccmmend ganking unless the enemy is on low health, until you have ur ult. Once you have ur ult. gank whenever you please. after you get wriggle's you can probably get blue or red buff, but make sure you have smite. Then the rest of the game should follow as a normal jungler plays.

Attack Speed/survivability is the way you want to go. all your moves are based on AD so get some of that if possible. Also you want attack speed whenever possible because shyvana's moves are all based on basic attacks. i.e. Burnout is extended by basic attacks, Twin Bite's cooldown is reduced, etc. Most importantly just be sure to get an item that slows ( Trinity Force or Frozen Mallet) and Sunfire Aegis. get Flame Breath first so you can check bushes at the start without having to go in them. also good for early game harass.

Just, phantom dancer, bloodthirster, infinity edge, phantom dancer, phantom dancer. 'Nuff said. ONLY TO BE USED IN GAMES WHEN YOUR ENEMY IS INSANELY STUPID.

when you are laning with a support character, generally you will have a shield at the ready, and maybe a heal depending on the support, and probably good harassment for early game. In this case, just go full offensive because you don't really need the extra defense/health due to your support. Always cover your weaknesses if you have a support though, so sometimes you might be against someone like Amumu who has AoE moves, then you might want to get something like mercury treads, etc. Always change according to the enemy and what support you lane with.