First of all:
I think there is no guide that works always perfect -
it depends always on the enemy team, the own amount of gold,
your own teammates ect.

Shyvana is very agile, can tank lots of damage and can also deal great DPS. You can almost always escape or chase and you can be very useful for your Team.
Since she has no CC besides the knockback from her ultimate, she is not that good for ganks as others; She also has a very bad early game when laning against ranged and depends on her items.

This guide is not supposed to be for the most damage or the best sustain,
but to be usefull in a team and to stay alive long enough to deal good amounts of damage.
To get this guide to work and to kick ***, you need at least a Warmogs, Wits End and the Boots - so just get as fast money as you can.

Twin Bite
Very usefull skill and the main reason for the trinity force.
Also, it resets the attack timer, so you can atack, press Q and instantly double attack again.
Maxed out, with a trinity force and the trick above, it brings you 4 times your normal autoattack damage, 6 rage, 3 times the effect of Wits End and so on.
Burnout
This skill is just great and gives you the ability to chase, to flee or just to farm as hell.
Max it first and almost noone will get away from you on the early levels.
Also try to use normal attacks and Twin Bite to increase its duration.
Flame Breath
Not that useful on the early levels, so just spend one point on it to get the armor-debuff.
In a fight, use it when getting close to an enemy to shrink his armor and deal some more damage. It is also helpful to poke an enemy if he trys to teleport back, check bushes and stuff like that.
Dragon's Descent
One of the most destructive and useful abilities in the game, not because of its quite good damage output, but because of the ability to escape and disperse groups of enemys.
In teamfights, use it from the sides - or even better - from behind to knock the enemys towards your team and seperate them a bit. Since you can take lots of damage with the early Warmogs and the utlimates passive, don't just run away - stay in the middle of the enemys, use her abilities and donate as much chaos as you can. Also try to autoattack / Twin Bite as much as you can, refreshing your Twin Bite-Cooldown and extend the duration of your Burnout. When you notice that you still have a problem, just use Burnout to evade and let your team do the rest - even when you die, they should be able to clean up the rests.
Besides this, you can use it to "Flash" over walls, chanse enemys ect. - don't be stingy to use it, you can stack up fast enough fury again with the AS from your Wits Ende, Zeal / Trinity Force and the Twin Bite.

I prefer Ignite and Flash, but the most Summoner Spells are usefull on her.

Recommended:

Ignite gives you a bit more damage burst and makes it even harder for enemies to escape.

Together with your ultimate, then tenacity from your boots and your Burnout, it gets almost impossible not to escape a gank. It also can be usefull to escape when you fail a dive into your enemies with your ultimate.

Useful:

You can chase / escape even better, but i think that Burnout is enough.

Very nice to counter AD or just prevent someone to flee, but for that you have your Burnout.

Okay if you're doing solo or want to gank a lot, but then I'd prefer Flash / Ignite anyway...

Rest:

Jungle-Shyvana is woking well, but your ganking is not that good till lvl 6.

Clairvoyance is quite usefull, but not on her. You have just one short-range skillshot to hit and shouldn't be that afraid of ganks.

You have Tenacity.

Tenacity, much HP, good base Resistances, MS-Buff, Ultimate to fly over terrain - you won't die often.
Unless you want to feed more quickly, don't take it. With no champion. Never. Seriously.

They see me trollin...

Get on a lange with someone ranged, don't take the mid - you're to squishy to get last hits or even harras the enemie. You can try to do a solo lane, but in this case, you have to play very defensive and should take teleport with you.

In early game, stay behind if your enemys have long ranges - for e.g. to lane against an Urgot can be a nightmare.
If your enemys are meele / short range, try to outzone them - just run behind their casters, active Burnout and run away if you get too much damage. But be careful, don't try this if your enemys have much CC or your teammate is a troll.
Farm as much as you can, use your abilities often and do as much normal attacks as possible - amazing passive...

In mid game, try to gank as soon as you have your Warmogs and your Ultimate.
Don't be afraid to dive someone if you have enough HP and a capable teammate - with your resistances from your ultimates passive, Burnout, Warmogs and your Ultimate, you can survive almost everyting.

When you have your Warmogs / the parts of it, you can tank the Dragon for a good amount of time - together with a DD, you should be able to get an easy dragon.

If it comes to teamfights, try to breach in from the sides / from behind with your ultimate, seperating your enemies, donating chaos and get in range for your AOE-Damage. Your teammates will love you for that if you're doing it right.

Start with a Regrowth Pendant and a healpotion to get very good sustain and to get your Warmogs more early. Boots and Healpotions are also a good choice.
I don't like Dorans Shield on her because you have enough HP and Armor with the runes, also it takes bit longer then to get the Warmogs ready.

As first items, grab the Mercurys Treads - the tenacity makes you unattractive to CC and the 25 mres are just great against any casters.

After you have your boots ready, get the Warmogs as quick as possible - as soon as you reach lvl 6, you will have high resistances and just need some hp to get very tanky. Try to get it full stacked quickly.

Wits End is just perfect for you, giving you more mres, AS for your passive and an nice addition magic damage, working with your Twin Bite.

The next items you should get are Zeal for even more AS / MS and Sheen for a nice damage bonus on your Twin Bite. If you have to deal with much physical damage, get an Atmas before finishing the Trinity Force - this gives you even more Armor and a nice damage increase.

The Trinity Force is just made for you: Some HP, a Slow, nice damage increase on Twin Bite and even more MS / AS.

A Hextech Gunblade is something you should get at last, only buy it if your enemy can't really harm you - if you still get too much damage, grab another Warmogs / Thornmail / Force of Nature / Guardian Angle, depending on your enemy team.

Items you shouldn't buy:

Guinsoo's: You can't keep the stacks up for a longer time, also AP is not that relevant with Shyvana.
Infinity Edge: Very expensive, gives you no sustain / AS, you don't depend on crits.
Rabadon's Deathcap: Your abilities and your passive only become effective over time,
so it is better to get some sustain or AS instead of a slight damage increase.

With this guide, you should be able to be useful to your team - or at least don't die that often.
Some stats of mine:
This is my first guide, feedback is welcome.
Also, sorry for my just moderate english.