My mom is Jewish and religiously, I was raised Jewish. However, my dad's side of the family is Christian and we'd celebrate with them as well. It was strictly secular, but we'd go to my paternal grandparents' house for the holiday, help trim the tree, open presents, etc. My husband, while not particularly religious, was raised in a Christian household. So, we still celebrate both as I did when growing up. The big difference is that I decorate our house for both. Growing up, we never had Christmas anything in our home because my maternal grandparents wouldn't approve. And yes, my kids get gifts for both. Most times, we give bigger on Christmas, but Hanukkah has been known to have some big ones as well. At least this year, Hanukkah is early, so my house won't smell like latkes once Christmas is upon us.
As for New Year's, we do celebrate January 1st. It's funny, because in Judaism, there are technically several new year markers, but Rosh Hashanah is traditionally looked upon as the major one. It's also when the Hebrew calendar moves to the new year (it's currently 5784). Some also look at a holiday called Simchat Torah (held on the last day of the festival of Sukkot) as another new year, as it signals the end of the Torah reading cycle and the beginning of the next (if I'm remembering correctly). Although, I don't think this is considered one of the four "official" new years. Passover, Tu B'Shevat, and Elul are the other official ones after Rosh Hashanah. I had to look that up because Passover was the only one I personally remembered being viewed as a new year beyond Rosh Hashanah.
Side note- Hanukkah actually starts at sundown on December 7th. Some sites will list the 8th because it's the first full calendar day, but Judaic holidays typically run from sundown to sundown. It does end on the 15th, but the last candle is lit at sundown on the 14th. So, with the sundown to sundown approach, it's over when the sun sets on the 15th.
