Hi Callie,
Welcome to KittyCatS !
Your cat will choose a partner on her own if you own other cats in breeding age around.
Once 2 cats have the breeding stats and are in range of eachother (and opposite gender of course)
they will mate and the female will give you a box.
You can also partner them if you want to choose who they mate with, same if you decide
to breed your cat with someone else's, cats will have to be partnered and both owners present
on the sim to agree on the partnership.
I don't know much about stud fees, but I am pretty sure you can find someone who will let
your kitty mate with theirs for free. Maybe twice, that way both of you get a kitten.
Don't hesitate if you have other questions.
Nen