Depending on your flexibility, meaning be able to accept a variety of neighborhoods and schools, the search could take as little as 10 days or as long as 6 weeks. I would allow about 30 days prior to your intended move-in.
You might find that the move-out of your current property is a critical item. While most properties are posted for availability 30 days prior, some apartments require more than 30 days notice of your move-out. This means you might be stuck with a period of time where you have given notice to move out, but you still haven't found a place to move into. Read your lease for notice of termination.
Some leases have some really awful provisions for early termination, and you might have to pay an extra month's rent (overlapping leases).
So, at least 30-40 days prior to your move, but longer if necessary to give notice on your current place or if you're picky about the new place.