There are a few things i should point out first:
1. For the sake of helping those too lazy to read the entire guide, i chose to mark the things that are best suited for general gameplay in classic mode. For those of you who ARE going to read on, take note that the above tables and charts may not show the exact setup that you are best suited to.
2. Shyvana is an all around good champion, or a "jack of all trades" as one might say, but she is the master of nothing, so be prepared to play a multitude of positions with her, but these positions will change in different games and Shyvana shouldn't be used as a replacement for one position (unless a teammate happens to leave the game or something).

Pros:
-good at everything except ranged
-very powerful and hard to kill late game
-good jungler early game
-she can turn into a DRAGON! :D
-really cool looking ;)

Cons:
-expensive build, probably wont finish the item list in 3v3 matches
-no natural CC moves
-not really good at playing roles like "tank" and "carry" she is mostly good for general play

We will start the full breakdown with summoner spells as the masteries and runes and items are all assuming you will be high enough level/good enough to get them.

In my opinion, Flash is a must. There are a multitude of reasons why flash is a must:
1. great for escaping
2. good way to catch fleeing champions (trust me, once you get good as shyvana they will start fleeing the instant they see you in their lane lol, well maybe not that soon, but after a few hits they tend to flee pretty quickly)
3. good way to get from lane to lane fast (altho this isn't the recommended use, if you are playing 5v5 you might find yourself hopping lanes a lot)

Now that we have established flash as the first summoner spell, what is the second? Well that has to be either Heal, Teleport, or Cleanse. Here is the breakdown of when to use which:

Heal: Use this when you are going in a partner lane (as i mentioned, lane hopping is somewhat common with shyvana depending on what playstyle you have, but if you have a partner in general, go with heal)

Teleport: Use this when you are going in a solo lane. It's good for solo laning because with shyvana your going to want to get back into the action fast because you have no one to hold off the enemy from advancing

Cleanse: This one is a risky choice. In draft pick games it may be easier to decide whether or not to use this (you'll understand why when i explain the next part), but if you're doing a blind pick then this will be a risky move. Cleanse is very effective if you are playing an offensive role, like pusher, but if there is no one on the other team with CC abilities, then what use is it? MOST of the time there will be a good CC champion on the other team, but you also never know if that champion will be your lane enemy. So Cleanse is a risky choice, sometimes it will pay off, other times you will never end up using it. Choose Wisely.

Items are the next big things, seeing as these are the next things that you fully achieve before summoner rank 30. The items are perhaps the most important thing that contributes to your success, because in pub matches, even if you are inexperienced with using shyvana, as long as you're using the right items you still stand a pretty good chance of winning. Items however are the most variable, as in, there are several items on the list that you will have to decide which is best for your current game. Here's the list in order of what to buy first through last:

1. Amplifying Tome: This is generally my starter item (either this or Doran's Shield when i feel like playing defensively, but normally i only play defensively when i have decent evidence that our team will probably lose), the reason being that my other items don't provide me with too much of an ability power boost. The boost isn't absolutely necessary, but very nice to have early game because of your lack of initial damage

2. Berserker's Greaves: This will initiate you into your main item build. I say this because your main item build will consist of attack oriented items.

3. Stark's Fervor OR Wit's End: Both decently cheap by this point in the game, or at least cheap enough to allow you to buy your fourth, and perhaps most essential, item. When should you choose one over the other? Use Stark's Fervor when you are playing with teammates, and Wit's End when you are in a solo lane. This is where i stop explaining my choices and you just trust that i am leading you in the right direction...lol.

4. Warmog's Armor (When you are playing on the defensive...in other words: you are being pushed back to your turret constantly), OR Sunfire Cape (When you are pushing and playing offensively), OR Force of Nature (When you are playing against a mage/spellcaster/person dealing tons of magic damage...Generally you don't want to get stuck fighting a ranged champion unless they are decently squishy, but for those times when you are...).

5. Hextech Gunblade (When you are quite wealthy...or just doing really well but need an extra push to launch you into a win), OR The Bloodthirster (When you are poor).

6. Thornmail, OR Madred's Bloodrazor. No real reason to choose either, if you have already got the other 5 items you have probably already won the game...but if you're still fighting either one will do. Note that you may have to sell the Tome (first item) to make room for building this item, but the Tome is pretty cheap and you can just buy it back after.

SUMMARY: Overall this is a good setup that you can use for almost any balanced melee champion. The core component of this setup is the attack speed and life steal aspects of it. The combination of attack speed and life steal should lead to you winning 1v1 fair fights. If minions\turrets are involved you may have a bit more difficulty, but if you are going into a straight up 1v1 battle this setup will almost surely get you the win, it's also great when farming because of all the life steal you can lay waste to minions while never really losing any health due to their low damage output and the excessive healing of yourself.

Ah the masteries, the next biggest impact on your game below items. The masteries i chose for Shyvana are indeed interesting. Not many people think of a 21-2-7 setup, but i did :D. WELL NOW, let's break it down
OFFENSE:
Deadliness-2: for the sake of earning those 20 points in offense. (Plus with all that attack speed, chances are even better that you'll get a critical hit).

Archmage's Savvy-3: A little ability power won't hurt ;).

Sorcery-4: None of your abilities apart from the ultimate use any form of cost, so you can cast them as much as you want, leaving cooldown reductions as a brilliant idea :D.

Alacrity-4: MORE ATTACK SPEED!

Sunder-3: More or less an attack boost.

Offensive Mastery-1: Good for farming.

Brute Force-3: Who doesn't need a little brute force every now and then?

Havoc-1: Good overall boosts.

DEFENSE:
Hardiness-2: I don't see you finding anywhere better to put it! (lol, jk, it is good for those games where u are losing).

UTILITY:
Good Hands-3: I don't know about you but i like getting right back into the action.

Perserverence-1: For the sake of moving down the ladder.

Awareness-3: XP is GOOOOOOD.

AND THERE YA HAVE IT FOLKS, that is mastery trees for beginners.

Runes...BLEH, who needs em, you only get these tiny little bonuses from em, but whatever still worth it i guess, if you have IP to spare. Well...not much to explain about them, just small boosts that relate to what i've explained earlier, so...you can really use whatever runes you'd like but i recorded the ones that i find luck with.

Here's the general pattern with the skill sequence:
Start with flame breath early game that way if someone goes for first blood against you, you can lower their armor and that will probably be a good head start to beat them early game, next your gonna wanna max burnout about as soon as possible, while still allowing room for 1 or 2 upgrades in twin bite and flame breath that way you have a well balanced champion, but you want to max burnout first so that you can farm and rack up the XP and gold so you can get your items, because Shyvana is not a very cheap character item-wise. That's about it, the skill sequence i recorded is just an example of a good one, but if your up against a ranged character you might wanna max out flame breath after burnout or in a melee lane you might wanna max out twin bite after burnout, just follow what you feel is right.