The Continental Divide (CDT) Trail can be accessed from several points. A common starting point can be accessed by taking Highway 180 out of Silver City, toward Gila/Cliff, New Mexico. Follow 180 out of Silver City until you reach Saddle Rock Canyon Road. Continue along the road for several miles until you intersect the Continental Divide singletrack. The trail is marked by characteristic blue placards.
From this northern starting point, you can ride the CDT down and through the Burro Mountains until you reach Highway 90, where the CDT can be taken all the way down to Lordsburg, Gila, and eventually its terminus at the US/Mexico border.
The CDT through the Burro Mountains is an incredibly robust ride of over 40 miles and is frequently ridden in segments by people who want shorter rides. The trails, particularly closer to Jack's Peak, offer beautiful vistas. However, this is a very dry and remote area, so plan accordingly and avoid riding this trail in the daytime during the summer months as the heat can be dangerous to the unprepared biker or the biker not used to hot, dry climates.
Hit up the folks at Gila Hike & Bike in Silver City for additional tips/directions. There are frequently group rides on the weekends from Silver City that ride in the Burro Mountains during the spring and fall and the bike shop may be able to get you in touch with these people if you are looking to join a group.