Hi. I have: Uncommon and Rare spices. But no Common. Once I get a common, can I just put in my indgreedients and tada a sylestie? Or, do they have to be certain ones?

Well, first off "I have: Uncommon and Rare spices. But no Common." is a statement that is, in fact, impossible, because Common and Uncommon pets cannot produce Special Spice so far as we are aware- only Rare or higher can.

What items you need to create a pet via baking varies from pet to pet, but is summed up as:
You need Baking Utensils
You need a Sylesti Spice of the desired species (obtained via using a Spice extractor on a Rare (blue star), Epic (purple star), or Legendary (yellow star) pet of the species, or buying/trading from other players)
You need the correct amount of both the uncommon and rare ingredients related TO that pet species (the amount required increases with the rarity of the pet.)

An overview of what pets require what and how much can be found HERE

Thus if you wanted to make a one-vis (Rare) Orange Cake Lupora, you would have to either find and spice a blue-star lupora, or otherwise obtain Rare Lupora spice, and aquire 18 Orange Slices (uncommon ingredient) 6 Shredded Coconuts (rare ingredient)
but if you wanted an epic Rosenmunnar Cookie Lunemara, you'd need to instead get an Epic Lunemara spice, 36 Powdered Sugars and 12 Jar of Jams.

But once you have everything, you just go to the Sylesti Bakery page, select the correct pet/ingredients and say 'bake'.

We aren't sure how useful we've been, but we hope we've helped.

That's very helpful! Thanks!