Hey!

In the following Chapters I will tell you a little about Items, Spells and also good Team Comps for Taliyah.
WARNING: I am German so don't mind the grammar mistakes. ^^
I hope I can help you improving your Taliyah-Skills.

So first thing you have to know: It is all up to you which items you chose. The first two items are the most important items but all the others depend on your matchup.
Some tips for taking the right items:

Archangel's is the item you definitely need first.
Morellonomicon inecreases your HP so I would take that as your second item.
Zhonya's is also an item which is in almost every Mage-build. Also a wise choice.
Rabadon's increases your AP by 120 AP AND your total AP by 40%. A good way to reache 700 AP...

As I already said, most of the items depend on your matchup or rather their items.
Just try to counter the items from your enemies.

Taliyah isn't as offensive like an Assassine. So the worst you can do is take Ignite. Here are a few options for your summoners:

Ghost: The spell I normally use because with that one you become even faster when you're moving near terrain/on "Worked Ground".
Cleanse: Take Cleanse when your enemies have a lot of CC that would make you die fast in team fights.
Barrier: A good choice against Ultimates which deal damage after a few seconds like Zed's R.
Exhaust: I would choose Exhaust as your spell against Yasuo because you reduce his damage by 40% and also his movement speed which is very important!

I think I don't have to say something about Flash because almost every champion uses Flash...

Wait until the minions stopped walking. Then use your E and try to damage all the creeps. Push the melee minions with your W right into the caster minions and then use Q to hit them all at the same time.
Do the same with cart minions as you do with melee minions.

The "Other Choices" are also a good option but Alistar is more tanky than Rakan.
Anyway this is a team with three very strong Tanks and a very strong Botlane.

I use the example team comp above.
You have three tanks for the frontline and four engage champions: Gnar, Zac, Taliyah and Alistar. So what you have to do:

While Gnar is raging and position yourself.
Right befor Gnar jumps in and becomes Mega-Gnar, Alistar goes all in and knocks them up and Zac jumps in with his E.
Taliyah says "We need to build a wall" (R)
Mega-Gnar can now use R to ult them against Taliyah's wall.
Alistar goes back to the backline (Tristana) and protects her while Tristana autoattacks/use E/Q/(R)
Zac uses R to jump with some enemies near Alistar and Tristana.
Taliyah stands behind the enemies and just uses E/W combo and spamms Q.

Taliyah's goal is to destroy the enemy backline and kill the ADC.
If you have low HP activate Ghost and kite them to death.