I'm a total newbie here, and when I was doing some battles in Fungus Grove to level up my pets, I randomly encountered a Rare Zolnixi. She's absolutely gorgeous, and I would hate to just kill her if there were any way to catch her. She's also level 20, the same level as my pets right now, so I could use her right away. The trouble is that I only have Lesser Snare Traps, which only catch Common and Uncommon pets. I can't find anywhere to get better traps. Is there any way for me to get this amazing pet without killing it?

You will need to follow the storyline until you've unlocked the second region, and have access to the sanctuary of saerielle. Until then, you can't get rare traps.

The only other alternative would be to wait for spring festival, if it comes out before you've unlocked the next region. During major festivals, in the festive zone, there'll be a merchant that sells uncommon, rare and epic traps, but with a limited supply that restocks every few hours. That's the only place you'll have access to all levels of traps sold in the different market, without needing to unlock their respective region.

So far, I don't think you have the ability to buy Standard Traps, the minimum tier traps for Rare Sylestis, unless you've reached the Sanctuary of Saerielle, in the next region, or in the Festival Zones of the festivals. So unless you've gotten that far, then sadly, no, you can't catch that Nixi. But there will be a lot more rare Sylestis when you've gotten further on, and you could always come back once you've gotten traps.