I am a Japanese but I don't think it is important. Sometimes Chinese characters of animals are used in names of places, though. But you don't need to learn them. I think firstly you should learn Kanji of 1st grade of elementary school like 日、上、下、見る、学校、車、雨、男、女、子　etc. For your information, 動物の漢字 of 1st grade is only 犬。

Did you learn 500?! That's great! It's time you started to learn Chinese characters of animals, then.

どこまでプロフェッショナルになるのかによりますが、 動物の漢字は特殊な場合も多いので、 一般的には先に回答にあるような 基本的なもの｛犬｝や｛猫｝だけで構わないと思います。

The Chinese character used for daily life is to read especially. At least, I think that I only have to remember these. (It only has to be able to read. ) 猫(ne ko)→cat 犬(i nu)→dog 牛(u shi)→cow 豚(bu ta)→pig 鳥（to ri）→bird

学习与动物有关的汉字是很重要的吗？動物の漢字——宾语；は——助词，提示主题；ですか——表疑问； 回答时，一般用： ——はい、そうです。 or ——いいえ、ちがいます。