Hospital will give you forms to do the following
register the babys birth
add them to your medicare card
any centrelink/family assistance payment forms you want to apply for e.g. baby bonus, family tax benefits a and b, parenting payment, rent assistance etc

by freelance i assume he is a sole trader with his own ABN? therefore they will prob want his last financial notice of assessment/BAS statements (to prove his income/expenses), and all his and ur usual ID, bank statements blah blah...
can i suggest if his pay is erratic your better off applying for family tax benefits at the end of financial yr, rather than opting for the fortnightly payments, cause u may find after your partner does his tax he may have earnt more than you assumed and you will have to pay centrelink back (cause how much family tax benefits u get depends on your combined incomes)
GL