I am a lover of Zyra so that's where my input will be coming from.
Zyra can definitely become bursty and she has... ok disengage, but she really thrives at creating engages IMHO. Your level 2 is fucking terrifying and you should use that to your advantage and chunk the enemy ADC with your ADC helping. Also, you can have epic bush control with your seeds and plants - do not forget it.
In regards to spellthief.. you'll naturally get it as a bonus with Zyra because of your plants or well-timed autos. I never make it a priority, it just happens for me. I don't play passive with Zyra unless I'm holding lane 1v2, or if I'm half health with no pots in which case I just play more carefully. I wouldn't be happy at all with 2 pots for 1 ADC pot - I would try to make that 1 for 1 at the most. Again, abuse your level 2.
Zyra is awesome at teamfighting. Try to grab an advantageous root - if you hit more than 1, great. Drop 2 Q plants then ult afterwards and you can do an epic amount of damage with or without your team. Respect your cooldowns, however, because you are immobile and squishy as all hell and you will die in half a second with no root. Fall back a little after you blow everything, but close enough to re-engage if possible.
As Zyra, I never upgrade the initial spellthief item. I keep it for a while but end up selling it off. Sightstone first is great, haunting guise is great, but you must upgrade it to Liandry's to become a monster. Your damage is stupid with a Liandry's. I like lucidity boots on her, but obv you can get sorcs or mobis if you wish. Don't just poke in lane, engage. Get that root and chunk half their health and make them scared of you.
When I'm laning, I watch to see if the enemy has bought any wards or a sightstone. If they have, I watch when they leave lane (tab) to see their items when they return. If they're missing a ward, or they're 2 charges down on their sightstone, you have an idea of where they warded based on how long they were gone and know where to sweep.
Pinks are amazing. I like to leave deep pinks if possible. I only use them as ward removers if I have the upgraded totem for pink wards.