On one self-teaching site, I learned to say "to not like" in a format like this: いぬはすきじゃりません。But in a self-teaching book it said to format it like this: いぬはすきではありません。I know there are many different ways to say everything in Japanese, but I just want to make sure that both of these are correct! What's the difference between the two? Is one more formal than the other? Are there certain situations in which one is more appropriate than the other?
is a casual colloquial expression.
is also a casual colloquial expression, even a little bit polite.
You can say and write the expression with "じゃ" in your e-mail to your friends and family members.
is a polite expression.
Anyway conversation topics of "likes-dislikes" are personal affirs, so you have many occasions to say "ｘｘｘは好きじゃない（=好きじゃありません）" with your friends.
じゃ is simply a contraction of では, so these sentences are equivalent and correct. じゃ is slightly more familiar/informal than では, but not necessarily rude or anything. You could probably functionally compare it to