My two Sylesti are both Level 20, and so are Talon's two Vulnyx. I keep getting defeated every time I battle them. Am I doing anything wrong? Do my Sylestis need to be at a higher level?

It's not really a question of level so much as your pets' setups. Getting through the game is kind of a min-max fest.

Flute made a really good guide to read up on about statting and gearing pets.

One quote you should definitely keep in mind when choosing relics is this: Tip: "Always use a relic that has either physical damage or magical damage, not both...a pet can only use one type of attack, physical or magical, and that will be based on whether the physical ability or magical ability is stronger. So stats in the weaker one is worthless."
Savynn