As Graham said the Terms & Conditions are a pretty good place to start. In my experience, though, every card is different and, a lot of the time, the bonus points will come through well before the T&Cs would have you expect them to come through.
Every card is different. In the case of AmEx and NAB, my experience has been that the bonus points came through within 24-48 hours of the spend criteria being met. With Virgin Money, the points came through on statement day (from memory the same was true of the Qantas Mastercard).
So you can’t “budget” for the points to come through as soon as you meet the spend criteria, but you may be pleasantly surprised!
In my experience I’ve never come across any card that gives you points when you sign up, before you spend anything.