Hello and welcome! Hopefully this thread will give you some the information on breeding your Sylesti.

1 - Themed
a) I think it's very important for us to talk about themed Sylesti. Themed Sylesti are essentially Sylesti with specific coloring. These Sylesti go for a lot more money than normal Sylesti, otherwise known as 'non-themed' Sylesti. These guys have a 100% chance of creating another themed pet IF they are bred with a pet of the same theme. For example...
I have a lot of Haunted Depths Ryoris. If I were to get a female Haunted Depths Ryori and a Male Haunted Depths Ryori, the egg would be a Haunted Depths Ryori.
If I were to breed a female Haunted Depths with a male Raspberry Cream Eyori, the result would not be a Themed Ryori.
b) Themed Sylesti can be affected by Pet Dye, and they will still be themed. This is called 'Pet Swing'.

2 - Why you should breed
a) One of the most popular ways to get pets is by breeding. As of time of writing, 4 million of the 5 million pets on Sylestia have been bred. Wow. So clearly that is the easiest way to get pets.
b) Another reason you should breed is the cost of breeding. Breeding is completely free if you have 2 compatible Sylesti. If you have a, say, Female Raspberry Cream Ryori, and you want an egg, simply head over to advanced search and select the filters. When I am breeding I usually select the gender, I include purebred, I input the traits I want, and last but CERTAINLY not least, to find the cheapest options, I order by sale price ascending.