Red Rock is a great place just West of the city. This is one of the first places I visited when I moved here almost 6 years ago. What has really helped me explore more of the area is geocaching and a hiking meetup group. I would have never realized some of the great hikes there are in the area without the help of others.
Some of my favorite areas are the ones not crowded with tourists. Follow the loop around to the far west side where there is less parking in the lots and those are some of the best hikes. Icebox canyon, a must! Everywhere along the road the views are crazy and sometimes you may even see a few burros running around.
What I do have a problem with now is that you can't even walk or bike in for free anymore. So when I don't feel like paying or don't want to buy a new year pass I just hop over the Calico and do hiking over there. You can hike into Red Rock for free!