it will takes weeks to months to get indexed. after that you may see a few 'here and there' hits.. from there you either disappear from sight for months (the dreaded sandbox), or you'll have a steady flow of traffic.. or, something in between.. Getting LOTS of quality inbound links will help speed up indexing and increase your search results.. adding good content never hurts either..
I put a website up ~ 2 weeks ago and I'm receiving most of my traffic from Google.... I guess it all depends on what your website is about.