Learn the characters at the same time as the language?
There is a key difference between learning the language as a native speaker, and learning the language as a second language. Native speakers of all languages invariably learn speaking and listening skills first, and only start to learn reading and writing at school, after the age of about 3 or 4. Learning a language as a second language, reading and writing have more prominence and can expedite some of the learning, but this also depends on a person's 'learning style'. Generally, combining all four communication skills is better.

To clarify what I am trying to say, my own experience of learning Chinese (and other languages) mixes all four modalities (reading, writing, speaking, listening). I find that learning the characters sometimes gives insight into the relationships between words (the semantics), that are not obvious from how they are said. Sometimes the process works the other way, as I see patterns in the written characters that are reflected in the sound.