This guide is meant to serve as a quick refference for any potential Shyvana players. The cookie-cutter build explains things well enough for you to make the best choices without having much thought.

TL;DR: Shyvana is a fighter-tank melee champion. She is resourceless, only needing Fury to activate her ultimate. Shyvana has great consistent hybrid damage while being innately tanky.

Her toolkit synergizes with clear speed, making her a potential counter-jungler as well as a lane pusher.

Shyvana deals hybrid damage, meaning she is difficult to itemize against.

Shyvana has innate resistances that scale as she levels up. Her ultimate doubles these resistances. This makes her a well-rounded fighter and potential tank.

Pros:

Hybrid damage.
Ultimate Knockback.
Innate tankiness.
Great clear speed.

Contras:

No CC besides ultimate. (unreliable)
Expensive item dependent.
Prone to kiting.
Melee range abilities with one ranged "nuke".

Early game:

Shyvana doesn't have a great start in the jungle, making her easy to invade. Try to have a safe start and head to your other buff to secure it and gank.

Top lane is the easiest lane to gank, try to gank there first. Mid lane should only be ganked if enemy has over-extended. Bot lane is lane-gank only, river/tribush ganks are not effective.
Use Exhaust to secure a kill only.

Farm in-between ganks, Shyvana is great at that. Try to have enough for your core before you back, else take Tiamat components.

Middle game:

Try to secure dragon. By now you should have Ravenous Hydra and basic tier boots. This is a great powerspike for Shyvana, making her a great skirmisher, abuse it if possible.

Gather with your laners to take down towers and dragons. Focus on the middle lane tower. Engage with your ult only if your teammates can follow up with crowd control. Do not dive right in to the enemy team.

Shyvana is a fighter, not a peeler. She is a hybrid of front-line and middle-line, so try to focus whoever is in range, prioritizing other fighers and the ocassional overextended carry.

Late game:

Shyvana falls of slightly, compensating by being a great offtank. By now you should have around 7000 effective health or so, making you a safe punch-soaker.
At this stage of the game you should make sure the enemy doesn't take Baron while you try to take inhibitors. Now you can dive right into the enemy team without much worry.
Stick with your team at all times, splitpushing only when necessary. Try to take as much damage and CC as possible, creating openings for your carries to deal damage.
Focus remains the same. Protect your carry if he's in trouble, though this applies always.

Shyvana is not a very versatile champion. Despite dealing great hybrid damage, she can only benefit from a thin selection of weaponry. All items that fit her well are included in the build section. Some information:

Offensive Items:

Ravenous Hydra - Core item for this Shyvana playstyle. Her auto-attack reset applies the passive, synergyzing with the physical damage it grants. Fast clear speed and great split-push potential.

Blade of the Ruined King - Helps Shyvana stick to her target. Passive works well with Shyvana's E. I prefer Hydra for the burst potential. This choice is better if you don't have trouble sticking to anyone.

Trinity Force - All stats besides mana benefit Shyvana greatly. No real downsides other than the price. Rush only if ahead.

Defensive Items:

Spirit Visage - Best magic resist choice for Shyvana. Passive boosts your Lifesteal and all the health regeneration you already have.

Randuin's Omen - Best armor choice. Active is great when diving with ultimate. Passive is sort of decent, helps against auto-attack based champions.

Guardian Angel - This is core no matter what. My Shyvana playstyle depends on middle and late game diving, making the passive of this item absolutely necessary. Rush it if your team depends on your ultimate for initiation.

Hybrid Items:

Maw of Malmortius - This item works best against sustained magical damage. The shield comes in handy against nukes aswell. The passive damage gain is great for a Shyvana that is always taking damage.

Frozen Mallet - This item works best as pseudo-CC. The passive helps us stick to a target by slowing it, helping our team to catch it easily. The attack damage and passive are the only reasons this isn't a defensive item.

Wit's End - Incredibly cost-effective, this item is best as a dueling choice against magical damage. It also synergizes with Shyvana's magical damage output, making the passive worth it.

Sunfire Aegis - An offensive version of Randuins Omen. This isn't recommended against enemies that can kite you, negating the passive. I prefer Omen, but this is still a decent pick.

As always, boots are situational:
Ninja Tabi - Heavy autoattack damage.
Mercury Threads - Crowd control.
Mobility Boots - Easier ganking and initiation.
Boots of Swiftness - Constant/Persistent slows.
Sorcerer's Shoes - Not recommended.
Ionian Boots of Lucidity - Not recommended.
Berserker's Greaves - Not recommended.