Learn Spanish (Spain): A ti te gustan los gatos, pero a mí me gustan los perros.

With verbs like gustar, the thing you like is actually the subject of the sentence. Since los gatos (cats) and los perros (dogs) are plural, the verb must also be plural (gustan). It is helpful to think of the sentence structure as cats are pleasing to you.

Are A ti and a mí required here, or can I just use te and me?

The sentence would be grammatically correct without them, but A ti and a mí are added here for emphasis. Because the sentence compares two different opinions, these extra phrases act like a spotlight to contrast the two people: You like cats, but I like dogs.

Why do we need los before gatos and perros? English just uses cats and dogs.

In Spanish, when you talk about a noun in a general sense (like the entire species of cats or dogs), you must include the definite article (el, la, los, or las).

Why does mí have an accent mark but ti does not?

The word mí (me) has a written accent mark to tell it apart from the possessive word mi (my). The word ti does not need an accent mark because there is no other word spelled that way to confuse it with.