They should have enough satellites in the sky, at the right positions, to start offering limited service around the start of July. This is based off of Elon's statement that they need 6 launches worth of satellites, educated guesses on launch dates, assuming that satellites need to move into their operational positions after that 6th launch, and will take the same amount of time to do so as previous launches.
They hope to be ready to help out with emergency service connectivity this hurricane season, which starts roughly speaking in August. That gives them around a months buffer, which sounds reasonable.
They don't seem to yet be making huge moves to get ground stations up and running, advertising to consumers, etc. This leads me to believe that the initial service will not be consumer oriented (probably as they work out the kinks). We know they're interested in emergency services, and we know the military is interested, these are likely to be higher paying and more forgiving customers to start with since they aren't competing with traditional ISPs at all. For them, somewhat unreliable service is better than no service.