Start looking casually now just to get the idea of neighborhoods. In terms of seriously looking at and touring apartments, 4-6 weeks ahead is usually the sweet spot.

Providence has a lot of students, so June-August is a very competitive start time, and prices are pretty high. When it gets closer, you might want to check out individual realty websites rather than the big collection websites. I got lucky and found a nice apartment before it got posted to Zillow