My dad is Jewish and my Mom was Christian, so I was raised in a blended family. The extended family on each side was strong in their respective faiths.
Dad says that when he was little, they did have Santa Claus. Grandma felt that it was a secular tradition that had nothing to do with the Christian celebration, so she just incorporated it. There were three boys, each roughly two years apart. Their tradition was that each night of Hanukkah brought a progressively bigger gift. The first night was always gelt (chocolate coins wrapped in gold foil) and a dreidel (a sort of spinning top). The second night was usually something needed, like socks and underwear, and nights 3-8 were "fun" gifts. I understand in a lot of households, night 8 might be something like a video game system and nights 3-7 might be games or accessories for it. Since they also had Santa Claus, though, in Dad's family the 8th night "big gift" was more moderate--maybe a book or something in that price range. On Christmas morning, then, Santa Claus brought toys.
As a kid, I got the best of both worlds. My Jewish grandma didn't do Santa Claus for me, since I got that at home, so instead she did the traditional 8-night ramp up to a big present. My parents lit the candles every night, and I got a little something each night, but most of my presents came for Christmas.
As an adult, I still celebrate Hanukkah with my dad. Now it's mostly about the candle lighting, but we still exchange gelt on the first night and give each other a smallish present on the last night. It's just fun.
As for the Menorah lighting, it's supposed to be done at sundown, although in my experience that's rarely exact anymore. You start with one candle plus the shamus. The shamus is used to light the other candles. Each night you add another candle, always from right to left, because that's the way the Hebrew language is written. After you light the candles, there's a short blessing to recite. Any present exchanges are typically done after the blessing but before the candles burn down. As mdsoccermom mentioned, foods fried in oil, particularly latkes, are common. But there's not a huge ritual with lots of food like there is for Passover.
The above are just my personal experiences and observations in Conservative and Reform Jewish congregations. I would imagine that traditions vary widely.