April 1:
Comet Hale-Bopp reaches Perihelion (the point in its orbit at which it is closest to the Sun.)

Evening viewing of Hale-Bopp will remain good for at least a couple more weeks, even though the comet will be leaving the Sun behind, and beginning its travels to more distant reaches of the solar system. The Moon will be cooperating, by being visible in morning hours rather than in the evening.

April 3rd:
The Moon passes 4 degrees north of Jupiter at 3 AM, EST.

April 6th:
Your clocks "spring ahead", and you lose an hour of sleep (most of North America).

April 7th:
New Moon, at 7:02 AM, EDT.
Moon occults Saturn.
Moon occults Venus.

April 12:
International Astronomy Day.

April 19:

The Moon passes 4 degrees south of Mars, 2 AM, EDT.

April 20-22:
Peak of the Lyrids meteor shower. They are seen as if coming from the constellation Lyra, to the northeast, especially after midnight. This is not one of the best meteor showers, and it has to compete with the light of the waxing moon.

April 22:
Earth Day.
Full Moon, at 4:33 PM, EDT.