Mid Zyra: Start Q as it's your main damage besides your plants. I usually rush Luden's as my main damage item and get ROA second. If against a Zed or someone similar, rush Zhonya's. Zyra is known for her great poke, AOE damage, and CC ability, but as you know she's mega squishy so beware of it. That's why building tanky is great on her, just like Annie. Her root doesn't last as long compared to say Morg, but it roots more than one person, which Morg doesn't have. My combo is EWQW then R and if that doesn't delete your mid enemy laner than also add ignite. Also Rylai's is great on her as her plants proc the slow and that also gives her a form of soft CC along with her root. Morello's burn is also proced by the plants. That's all I can think of right now.
Support Zyra: Start E. Poke with your plants but always keep your E open in case a jungler ganks or the enemy bot laners try to go in on you. Get Frost Queen, Ruby Sightstone, and your choice of boots, and somewhere in that line start the items for ROA, and finish it after you get your support items. Then you can pick and choose what items to get, like Locket if against AP heavy, or Zeke's against AD. You can go for more damage too.