There are lots of great day hikes in the Red Rock area. Another great place for day hiking is Valley of Fire State Park. It's about an hour north of Las Vegas and you could easily spend a full day there.
Don't forget that Zion National Park is about 3 hours away in Southern Utah. Zion is worth a few days if you haven't been there before. Lots of great hikes there but it all depends on the weather.
Remember that the price of hotel rooms in Las Vegas almost triple around New Years. That might be the time to check out Zion. Before Christmas you are normally pretty good but as you get closer to New Years the hotel room prices keep rising.
There are camp grounds at Red Rock, Valley of Fire, and Lake Mead if you wanted to camp.
Let me know if you have any questions since I have lived in Las Vegas along time. Thanks.