Jupiter starts the month very low in the west just after sunset. On the 1st, it sets an hour after the Sun, but a week later, it will have essentially disappeared into the solar glare. It reaches solar conjunction, passing "behind" the Sun, on June 19. This happens roughly every 13 months, but because Jupiter's orbit is slightly inclined to the ecliptic, the planet usually passes slightly above or below the Sun as seen from Earth. Twice per Jupiter's 12-year orbit around the Sun, however, it actually gets occulted by the Sun; this is one of those occasions. Of course, none of this will be visible to us on Earth due to the intense solar glare. By the end of June, Jupiter begins to re-emerge into visibility in the morning sky.
Venus is slowly climbing higher in the west after sunset. Throughout June, it sets about an hour and a half after the Sun. It remains at magnitude -3.8 all month - dim for Venus but nevertheless brighter than any other planet ever gets. If you look at it through a telescope, you'll see an almost-full disk just 11 arc-seconds across. The planet will continue to get higher, brighter, and larger in the weeks and months ahead.
Mercury is undergoing its best evening apparition of the year for Northern Hemisphere observers. On June 1st, it stays up for an hour and three-quarters after sunset. On that day, Mercury, Venus, and Jupiter form a line, with Mercury on top and Jupiter at the bottom. The configuration doesn't last long, however, with Mercury sinking, Venus slowly rising, and Jupiter dropping out of sight. Mercury reaches greatest elongation (24° east of the Sun) on the 12th. The planet then shines at magnitude +0.6, and, in a telescope, presents a disk 8" across and 38% illuminated. On the 19th, Mercury and Venus pass the closest to each other, with Mercury less than 2° to the lower left of Venus. Compare the crescent disk of Mercury, 23% illuminated and 10" across, with the similarly-sized nearly-full disk of Venus.
Saturn is high up in the south at sunset, and opportunities to view it could hardly be better. It dims slightly this month, from magnitude +0.3 to +0.5. Its distance from us increases from 1.345 billion km (835 million miles) to 1.4 billion km (870 million miles). The disk of the planet itself spans about 18", while the visible rings are about 41" wide; the rings are tilted 17° to our line of sight.
By the end of June, Neptune is rising well before midnight. The magnitude 7.9 planet lies about 40 arc-minutes NW of Sigma Aquarii. In a telescope, it appears as a bluish disk just 2.3" across.
Uranus rises before 1 am by the end of June. It lies about 3.5° to the S of Delta Piscium. The magnitude 5.9 planet presents a 3.4"-wide grey-green disk in a telescope.
Towards the very end of June, Mars becomes visible very low in the eastern sky before dawn. On the 30th, it rises an hour and three-quarters before the Sun, but, at magnitude +1.4, it is rather dim and difficult to see against the bright glare of twilight.
Pluto, in Sagittarius, is approaching opposition, so is highest just before midnight. Still, at magnitude 14, it presents a challenging target even in a large telescope.
The dwarf planet/Main-Belt-asteroid 1 Ceres shines at magnitude 8.8. On June 6, it passes just 0.65° S of Pollux. On the 15th, it crosses the boundary from Gemini into Cancer.