Stars and Constellations
In December, the summer constellations are still visible for the first few hours of the night towards the west. The Summer Triangle is still fairly high, with Vega heading to the west-northwest, Deneb closest to the zenith, and Altair in the west-southwest. The center of the Milky Way is now below the horizon, but there is still a good portion of our galaxy we can see streaking high across the sky. The fall sky is now very prominent, getting high in the eastern sky. The "W" of Cassiopeia is very high in the north-northeast. In the absence of the Big Dipper (part of our spring sky) Cassiopeia can be used to locate the north star: The top (open side) of the "W" faces to the north, so in that direction look for a star about the same brightness as the main stars in Casssiopeia, and that will most likely be Polaris. Next, the Great Square of Pegasus is nearly in the middle of the sky. Andromeda is nearby in the northeast, with Perseus just below. There is also a fairly bright star called Fomalhaut in the south, though its constellation Piscis Austrinus is not easy to distinguish. Finally, the winter sky is beginning to come up along the eastern horizon. Taurus the bull is now up in the east with the bright star Aldebaran and the Pleiades (aka the seven sisters or Subaru) star cluster. Auriga the charioteer is also up in the northeast, with the bright star Capella.
In January, the summer constellations are now very low in the west. Deneb is the highest point of the Summer Triangle in the northwest, with Vega below near the horizon, and Altair more towards the west. The Milky Way continues to streak across the sky, though the summer portion is now giving way to the winter portion. The fall constellations are now right in the middle of the sky. More of the winter constellations are now up, most notably Orion the hunter in the east below Taurus. Gemini the twins are also up in the east-northeast just below Auriga
Solar System
Venus is still a bright "evening star" in the southwest after sunset, In the constellation of Capricornus. It is now a thin crescent shape if you view it through a telescope, but is now getting closer to the sun in the sky.
Jupiter is now coming up soon after sunset and can be found in the constellation Gemini.