A lot of people claim that Taliyah is Underpowered. I would say otherwise. She is what is called a "High Skill Floor" Champion. That is, she takes a higher level of skill and game knowledge to play at an acceptable level than many other champions.

This introduction is for Taliyah taken Support. At time of publication, the other guides are not done yet, but they will be soon.

Pros:

Insane Damages and scalings
Zone control
Synergizes at godlike levels with ADCs that have CC in their kit
Can function as a damage support, or an off-tank engager

Dat Ultimate (Her creator literally wonders why they let it ship)

Cons:

People think that she's terrible. You will get flak for playing her

Dat Ultimate
High skill floor
Single main CC spell
Long cooldowns early on

Your early lane is much like any other support. You take the AP poke support item, and Q first, unless your team is invading (in which case, take W first, Q second, getting E at level 3).

Poke the opponent whenever it's easy. Remember the small explosion radius on the q. If you can hit them by hitting minions, feel free as long as they have enough health to survive a small beating.

As a support, your job is to poke down the opponents early, reducing all-in potential before most engage supports get into their own. With Taliyah specifically, take advantage of the insane damage on your Q's full hit, as well as Thunderlord's on auto attacks. The opposing (melee) support moving towards your Siege Minion is a great time to make them regret life. Another important point: Two single stones at 30 mana each deal the same damage as 3/5 stones at 60 mana due to the 50% reduction after the first hit. using Q from Worked Ground can also keep your opponents off-guard, as well as giving you a movement speed buff for moving in-and-out very safely.

Once you have E and W, it's a whole different lane. With Jhin, MF, or Ashe, you can depend on your lane partner, being extremely reserved with your spells and waiting for your ADC to slow/root them while you fish with Threaded Volley for Spellthief's procs. When they land one, react quickly. Press E facing about halfway between the enemy ADC and Support. Now, look for positioning. If you can land the W for free on the ADC, throw them into your minefield, activating Q as they land. Throw some autoattacks in there, and you should actually be able to kill them from 100% to zero if your ADC has half of a brain. Remember, Exhaust can help both deal damage and reduce turnabout if they try it.
If you Can't hit the ADC into the minefield, try the support. If its a tank on high health, then you lost the E cast... but a knockaway W can ruin their plan to capitalize on your E being down, giving your side time to reposition.

If you can't depend on your lane partner for CC, you have to play slightly differently. If the enemy ADC has some sort of skillshot that takes a cast time (Lucian and Caitlyn come to mind), wait for it. When they cast it too close to you, cast your E at them, then Animation Cancel the E cast with the W cast, again activating Q as they land. Your full combo deals a lot of damage. Taliyah is a dangerous force, but her abilities must be landed completely in order to unlock her damage.

If mana is an issue, you are likely "fishing" with your W or your E. This means that you are wasting damage, and likely pushing your lane. Try to save those abilities more.

By the way, the reason the "Core" is aligned like it is is because you should buy Haunting Guise before starting Rylai's, then complete Rylai's before Liandry's. This leads to higher consistent damage of the whole course of building.

Oh boy. Taliyah's Ultimate. This little gamechanger is probably my favorite ultimate in the entire game (And I main Zyra and Ekko).

Treat it like a nuclear-f***ing-missile. A well-timed and justified use of it wins games. A badly-timed or unjustified use gets you killed. Over. And over. and Over again. This is why so many Taliyahs feed so hard. They take the ultimate when they just... can't. Or they take it to the end. Or they get off on the wrong side (It's literally just right-click off to get off that direction :/ ). Or they box off an ally and don't deactivate it. Yes, Taliyah can press R to take down the wall about a second after it is casted.

Roaming with this is your bread and butter. Its great for ganking midlane or toplane. Taking the ult, block off the escape route. Get off a little before the end, likely in the middle of the lane. E at your opponent as you close with Q active. W then from inside the E at your ally, and they should take the Hugely-chunked enemy and finish them off with a few autos or a spellcast.

If the enemy is mobile and looking to escape, hold the W cast. The E should dissuade most mobility spells within it, but as they go over the wall, prep your W in their target location. It can and will throw them over the wall, and you will often get a flash even from a mobile laner.

Remember, you don't always have to take the wall. If you're just warding drag and your jungler ganks mid, feel free to throw a no-commitment ult up there if you can't follow yourself because of your HP or planned pathing. If the opponent burns flash early, you secured a kill, and if they didn't, you probably burned a flash.

3 important things:

1. Keep moving: Taliyah's damage comes from a moving zone. Work like an avalanche or a mudslide. Endless Q slows and the occasional throw into an ally with CC or even just back into your team can rip apart even the most solid of team compositions. An E can dissuade more fragile divers, but when you have W, you can punish bruisers and assassins alike for thinking your ADC was vulnerable without a tank crouching on top of them.

2. Don't be afraid of wandering off a bit for the perfect Ultimate Angle. Being out of combat for just a couple of seconds and wandering off with your passive to land a team-splitting ult can turn a fight into a pick. If their damage dealers are trapped behind a wall, their tanks are free gold.

3. You are a squishy mage... NOT. You are a not-so-squishy mage. You aren't a tank, but for a mage, you can take a beating. Taking a little bit of damage from a diver by drawing them onto you can be a great way to bait out an enemy, so long as you have a quick Exhaust finger.

Be a tank. be a frontline. The ultimate is like Sion's, but stronger. Sure, they get warning... but being CC'ed, thrown into the enemy team, and having a tank in your face is still being CC'ed, thrown into the enemy team, and having a tank in your face.

Jokes aside, you engage REALLY WELL. Constant slows are still a thing you do. The W is a lot easier to cast on people you're basically co-positioned with, or who are still knocked aside by the ult's cast..

This isn't a build I use often, and it generally comes from a transfer because the game is going very long and we can use an off-tank more than a mage due to team composition. Do not build this if you're the only non-tank magic damage.