The promotion to Guardian requires you to complete two quests – “The Fugitives” and “The Wandering Scholar”.
The Fugitives – Talk to Burz gro-Khash and he will tell you about four escaped prisoners in Bravil that are creating trouble with the townspeople. Travel to Bravil and talk to the locals until you find out that the fugitives are hiding in Bloodmaybe Cave, northwest of Bravil. Explore the cave until you find and kill all four escaped prisoners. When they are all dead, return to Burz gro-Khash for your reward.
The Wandering Scholar – Talk to Azzan and he will instruct you to protect a scholar named Elante of Alinor who wants to study a Deadric shinre in Brittlerock Cave, northwest of Kvatch. Travel there to find her waiting outside the cave, but do NOT talk to her. Instead, enter the cave and kill all of the monsters that you find on the upper level. Once you’ve killed everything, return to the entrance, talk to Elante and follow her through the cleared cave to the Daedric shrine. Talk to her once you reach the shrine and she will send you away. Return to Azzan for your reward.