Learn them at the same time. thats what I do. first learn the pinyin. then jump into characters. many chinese words have the same sound and same tone but different meaning and different character. just looking at pinyin is confusing. However, looking at the character is not as confusing. my favorite is 肌肉 ji1 rou（muscle) and 鸡肉 ji1 rou4 (chicken meat). both have the same sound and tone but the character ji is different.

In my opinion, it's better to start with speaking and reading. Compared with English or Spanish, Chinese is easier to speak but more difficult to write. Chinese grammar is easy because you don't have to remember the tense of the verbs or the gender of the nouns. (For example, when you mention something happened in the past, you simply mention the point of the time and the verbs are always the same.) When you learn how to speak, you'll feel much interested and confident, then you can go further into reading and writing.

I think, just as all the langues in the world, alway the oral goes ahead of the writing :) But it really depends, you should judge from what you are gonna use your Chinese for :)

In my opinion.... Speaking always goes first..

I don't know writing here means write words or write articles ? If here it means write words, of course it is advisable. Or if it means write articles, maybe you can learn speaking first，and only when you known how to write the words can you learn writing