Hi Jess,
There are three steps in the process: sponsorship application (this is done by the employer), nomination application (also done by the employer) and visa application (this can be lodged by the employer or you).
From what you have written it looks like you only have to worry about the visa application part. So the employer will undoubtedly let you know when the nomination is approved and when you can apply for your visa.
The application will be lodged online, and all documents can be attached online (your ID documents, evidence of qualifications and your work experience). At the time that you apply online you will be expected to pay the visa application fee (via a credit card). This is currently AUD$455. The applications normally are processed quite quickly.
The rules with respect to the 457s will be changing on 1 July, and this seems (to us) to have slowed down processing at the moment. So you may need to be patient: the employer may have been asked for additional documents that they did not need to provide previously, and so on.
While you wait to hear from the employer, you should be getting together all the documents (have them scanned in) and maybe investigate your nearest approved medical clinic (this can be found on the immi.gov.au website).
Cheerio,
Veronika