The short version
Open when your town's daily average air temperature starts climbing through the low-to-mid 60s°F — water follows to swimmable within a couple of weeks. Earlier is safe (a circulating, chlorinated pool doesn't grow algae; a warm covered one does). Later costs real money: the classic green-swamp opening is almost always a pool that sat covered and warm too long.
For a single answer for your town, the pool season lookup takes a ZIP code, and every city page under pool seasons pairs the dates with a month-by-month warm-enough table.