The NCAR trailhead is a great place to start a hike. It puts you right in the middle of the Mesa Trail and provides access to a lot of great hikes either to the North or South. NCAR, the National Center for Atmospheric Research, has their beautiful IM Pei designed building at the end of Table Mesa Rd. Turn west from Broadway onto Table Mesa and climb to the top. There is a large parking lot, which makes this a great place to go on busy weekends.
From the NCAR building, there is one main trail to the west. Take this trail to join the Mesa Trail and continue on to other trails. One of my favorites is to to go south on the Mesa Trail and then peel off onto the Bear Canyon trail. This trail takes you through a high walled canyon with waterfalls down below in the spring. From the top of Bear Canyon, you can can take the Green or Bear mountain trails to their respective summits.
Another popular hike from NCAR is to Mallory Cave. It is a small cave, but worth the trip for the views. I will be adding more information and linking to write-ups of some of the hikes in the future.