Taliyahs summoner spells depends on how you're gonna play her.
Ignite for trying to kill enemy mid laner early.
Exhaust if you're vs someone like LeBlanc, Ahri, Zed, Talon, Yasuo
Flash Ghost if you're looking for early roams and looking to win mid game.
TP if you're in a losing matchup
Heal/Barrier if you're afraid of someone like Zed, LeBlanc and so on.

I always kill caster creeps level 1 to make myself more lane dominant and having the control over the lane. Taliyahs laning phase is extremely strong at the moment due to her q being able to kill all 3 caster creeps rank 1. Keep last hitting with your aa or Q in worked ground, and then Q out of worked ground to push lane. To push lane use e on melee creeps, W caster creeps onto E and use Q to kill the surviving cs.

Always get Lost Chapter on your first home, unless you've been forced back before 900 gold, if so, buy extra dorans ring, pink ward and refill potion (or regular pot, if you cant accord refill). Finish Morello asap, unless you're vs a burst mage like LeBlanc, Syndra, Ahri, then go Lost chapter into Abyssal. If you're ahead/winning lane vs said burst mage, go lost chapter>negatron cloak into morello, then get rylais into abyss. Always get rylais as your second item, without it you're gonna miss out on a lot of free kills and picks. As for your third item, you can either go deathcap if they don't have any significant magic resist, but if they do have magic resist, go void staff into deathcap. Your 6th item is very dependant on the enemy team, but Taliyahs kit is so good, almost any ap item can work to great extend on her.

Side note: As for boots, you generally wanna get Sorc as you rely on magic pen and flat ap, but I often go swifties if I have lost chapter and boots early on the game, this allows you to roam a lot more without having to use your ult, which got nerfed (cd) in 6.15. Taliyah almost instantly clear waves without a lot of ap, so it's easy to roam with her without having to use ult. Mercs is obvious, when you're vs a heavy cc team or you're getting smashed by strong ap user mid. Another note; I haven't really played vs Zed or Talon as Taliyah, so I don't know what you would wanna get, but I imagine Hourglass into Rylais would work.