The drive to Globe from Phoenix is awsome. The Boyce Thompson Arboretum (en route) is very worthwhile for a hike and lunch.
Getting to Sedona, as Tom D. notes, is easier (frankly) if you go back to Phoenix. But, that is hardly the way an Arizonan would do it.
The adventurous drive is NW from Globe on Hwy 88 and Hwy 188; then Hwy 87 to Payson. (The Rim Club is worth it).
From Payson, follow the Gen. Cook Trail (Hwy 260) to Camp Verde; then I-17 North and on to Sedona.
The Gen. Cook Trail is described thusly from a tourism source: "The Crook Trail is named after General George Crook, the Commander of the Military Department of Arizona in 1871, who was in charge of subduing the Apache Indians and confining them to reservations. General Crook, who was known as "Gray Wolf" by the Apaches, rode mules along the trail as they were more sure-footed than horses. The trail was built under Crook's command in 1872 to connect Fort Whipple, the Arizona military headquarters near Prescott and Fort Apache. Supplies and troops were moved to Fort Apache on this trail and for 22 years the trail was used as a route to patrol the northern boundary of the Apache Reservation."
Highway 260 does not exactly follow the trail, as I recall. But it is very cool land to see; forests, river valleys and American Indian country.
A side trip worth it is northeast from Globe on Hwy. 60, through the Salt River Canyon. This is often called the "Mini Grand Canyon". It is out of your way, unless you take your time and explore Arizona, the 48th State.