As a lower tier but frequent Zyra player, I might be able to help you.
Just being cautious will go a long way. As Lulu already said, Zyra's an excellent teamfighter, so if you're not winning lane, there's no rush. Her AOE damage and CC are just incredible.
Try using seeds sparingly until you go all in. I only use one seed when I'm just poking, and save the 2nd and recharges. If you have a second plant attacking the same target, the second plant's damage is reduced, so keep that in mind.
Your seeds give vision. Use them. They may not be wards, but they'll help. And if someone stamps them out you still have vision of them for a bit.
Don't underestimate your damage, especially as time goes on! If you're fighting another squishy, you can take them out! Pretty damn quickly if you chain the combo right, to. Even some squishier melees can melt against your AOE and plant damage later in the game.
Lastly, use your plants to max affect when pushing. If you're going B, throw out 2 plants at the minon wave and let them clear it for you while you return to base. You can also snag some jungle camps pretty quick, if your jungler is fine with it, of course.
That's all at the top of my head. But Zyra's really underrated IMO. She's worth to learn.
I think Lulu already covered ideal match ups pretty well. Shorter range, less bursty mages. Orianna, Cassiopeia, etc. are a few other good ones.
Good luck!