Flash is essential. You always take it. Chase enemies and Q/W/E them or get away from sticky situations. Teleport is nice. You can get away in eggform if you get the timing right, you get in lane fast, you get in objectives late game, you roam. You take Ignite if the enemy team has healers like Dr. Mundo/ Warwick/ Soraka and ur team takes no Ignite, or if you are against a Vladimir or any other heavy healer. You take Barrier when you want ot play more safely or if you are on a difficult machup or playing against an assasin. You take Cleanse if you have a heavy cc champ in lane like Veigar or Twisted Fate.
SPECIAL CONDITION: against Yasuo or Yone, always barrier or ignite(they can interrupt your tp easy af and cleanse does nothing to them)

Doran's Ring is just better than Tear of the Goddess, but it is really ur choice, since tear is a component for your core item and helps you stack faster. You take the Dark Seal start ONLY IF YOU ARE EXTREMELY CONFIDENT YOU WILL WIN IN LANE.

Take early boots because most champs in the game have skillshots. Potions are really helpful, they offer a large ammount of sustainability in lane. Always, always, ALWAYS take Control Ward.

This should be easy. Liandry's Torment if they are too tanky, Riftmaker if you are far ahead and Everfrost if its none of them.

Archangel's Staff is in a bad spot rn but its still essential for some champions. Anivia needs it because of the late game potential(max. mana and shield). The boot section is easy. If they have too much ad, Plated Steelcaps, if they have too much ap and cc, Mercury's Treads. If evenly splitted or if you are far ahead, Sorcerer's Shoes.

Void Staff if they build mr. Chemtech Putrifier if they have healing. Plain and simple.

Zhonya's Hourglass if they have more ad and/or burst. Banshee's Veil if they have more ap and /or poke.

This is a bit more complicated. You usually want to finish with a Rabadon's Deathcap. However, if your time doesnt understand that you have to stay protected in teamfights, or if they just have champions that go easily to the backline like yone, i would suggest Demonic Embrace. Horizon Focus on the other hand is a very interesting choice. I think its viable only if they have relatively ranked champions and only if you can hit skillshots easily from far away.

This chapter is crucial but easy. Your Q Flash Frost freezes people when you stun them. Your R Glacial Storm freezes people when they stay even a millisecond inside when its fully formed. Your E Frostbite does double damage to frozen targets. Your W Crystallizeblocks their way. You max ur E first for insane damage. You can choose to max W second, but it doesnt really worth it.

Fairly simple. Arcane Comet with Precision is the best and more common. It really depends on your playstyle. The Electrocute page gives you more damage but less mana stability.