I've noticed that some pets start off with higher base stats than others. How do you breed pets with high base stats? Does it depend on the base stats of the parents? Is there a lot of variation involved as well?

when breeding, the final stats of the offspring depend on 3 or 4 things

1. the base stats of the parents.
- for each offspring, a random number between the two parent stats will be picked
- example, parent 1 has 20 Str, parent 2 has 25 Str. All offspring will have 20, 21, 22, 23, 24, or 25 as their base Str

2. Assisted Hatching,
- for 24 hours after an eggling hits 72 maturity hours (after 24 hours it autohatches), the player can interact with it to affect the stats upon hatching.
- the three options are (paraphrased) "Help it hatch" (gives +20 to hp), "let it hatch on its own" (+1 randomly to one of Str, Int, Dex, or Agi. Already maxed stats will be removed from the pool of options if I remember right), and "Use Philter" (with a dropdown to select a philter such as the Lesser Philter of Vigor or Prismatic Philter from the Inventory to use. What affect this has on the stats will depend on the philter selected)

3. Nurturing
- while a sylesti is in the Hatchling stage, it will gain +1 to its lowest stat for every 8 correct nurtures it recieves. HP is counted slightly differently (since it maxes at 2k rather than 100), and a pet can only gain points in Mana this way once all other stats on a pet are maxed. (this is counterbalanced by the pet gaining +1 maturity hour for every 4 correct nurtures)
- especially potent during fest when more people are active and nurturing.
- Nurture boost can increase the stat gain by virtue of making it more likely that the sylesti will be nurtured (since it will appear at the front of the nurture lists), though it makes no actual difference to the way the nurturing functions

4. LG Bonus stats/ Wild bonus stats
- pets that are caught in the Lost Grove with a rarity uncommon or higher have these + stats added to their Base stats.
When they have an offspring, these are inherited with some loss. I have a bad memory, but I belive it's something like losing 60-80 HP, ~10-20 mana, and 3-4 from Str/int/dex/agi per generation separation from the original LG pet.
- If you breed two pets with LG stats, the same function as point 1 occurs, picking a random number between the two parent stats, THEN subtracting the generational loss
- if one parent has an LG bonus stat and the other parent doesn't, then it skips the first step and just subtracts the generational loss.
- Examples:
Pet 1 has +10 Str, Pet 2 has +12, Pet 3 has +0
Offspring of 1 and 2 woudl be anywhere between +6 and +9 (10-4 on the low end, 12-3 on the high end)
Whereas 1 and 3's offspring would be either +6 or +7 (10-4 low end, 10-3 high end)
- LG bonus stats cannot be altered at all

Like Scathreoite said there are a lot of options for hatching and catching pets to breed, some people will also used themed pets to make max stat and supermax stat pets and make use of Philters when hatching their eggs to increase their base stats.
When hatching an egg, if you help the egg hatch it will also receive a bonus to its health or mana, and if you choose the other option to the right of the menu it will increase its lowest stat (strength, dexterity, agility, intelligence, etc). There are also Philters available in the scale shop which can increase the base stats by 5, or a commonly used by project breeders, Prismatic Philters which give a bonus to all stats including health and mana. All Philters can only be used once on a pet while they're hatching though, so it will take a lot of generations. ^^

Got it, thanks!