The most notable event this month is one most people couldn't — except through a telescope — see, and that was the transit of Venus across the face of the Sun. It occurred on June 5 and won't occur again until 2117.
Hurray! Summer is here at last. The solstice occurs on June 20. Mars and Saturn (now both in Virgo) are our only evening "stars." Venus has left the evening sky after being a "beacon" all spring and Jupiter is too near the glare of the sun to be easily seen. But wait until next month.
Note the summer constellations coming into view. The "handle" of the Big Dipper (part of Ursa Major — the big bear) points towards Arcturus (in Bootes — the herdsman) and thence towards Spica (in Virgo — the maiden). Saturn is still near Spica.
The zodiac constellations this month include Libra (the scales), Virgo (the maiden) and Leo (the lion). Hydra (the water serpent) is spread across the southern sky. At least temperatures this month and next are conducive to night viewing!