Should be enabled by default on both Cisco switches. If not, then enable it on each VLAN that will be trafficing across your redundant connections.
If your UTP backup link is on a 10/100 port, it will never carry data until the 1000-megabit Gig Ethernet connection goes down. Spanning Tree should figure out that the Gig link is the preferred one.
When the Gig link goes down, there will be a stoppage of network traffic on the affected VLANs while the network reconverges. By default this will take from 30 to 50 seconds. You can shorten this interval by tuning the the STP timers, but you need to know everything about your network before you start doing that.
A better way to shorten the reconvergence interval is to enable a feature like Spanning Tree UplinkFast on one switch. This will reduce the time needed to 3-5 seconds.
IEEE 802.1W Rapid STP may also be an option to speed the failover time: it's supported on both of these switches.
