Get gain on your electric guitar. The "crunchy" gain sound on Audioslave tracks is not the extreme gain of "heavy metal," but there is a somewhat fuzzy sound to the riffs. Another guitar sound is the ethereal quality of the guitar work on tracks like "Like a Stone," facilitated by effects like vibrato or reverb.
Practice going through repetitive note structures like the repetitive rhythmic riff on Audioslave hit "Be Yourself." In tracks like this, as well as quieter tracks like "Doesn't Remind Me," the guitar acts as a frame for Cornell's repeating vocal phrases, again resolving a "C major" type low chord.
Use minimalism, sustaining notes on the guitar, and find the melodies of the Audioslave tracks. Audioslave enabled Morello to put down riffs on slower, more melodic songs, and the results are interesting, as in the ethereal guitar work on "Like a Stone." From the more minor-sounding two-note progressions over the intro to the wailing guitar solo, a different sound is present, a more wistful melody than you'd find in either of the "parent bands." There's also a kind of guitar minimalism displayed in a single-strike chord through the verses of the song.
Use tablature to "track" Audioslave solos. Tablature sheet music, or "tabs," is an efficient way to show the notation of guitar solos. Find Audioslave tabs online or in guitar magazines.
Utilize the "funk" quality of tracks on the last album "Revelations." The drop-downs in the title track and "Original Fire" (in the key of E, it would be a bent G to an E major) is one example of a much-used "funk" style riff. Another is the blurred-note quality of the solos on both tracks, with wide-range crescendos resolving to the low note of the dominant chord.