On this page of the game guide to Borderlands 2 you will find the walkthrough description of the main mission titled Hunting the Firehawk. You will complete this task while progressing through the Frostburn Canyon location.
1) Starting point/teleport to Three Horns Divide
2) Fork in the road
Go up the stairs to the first floor of Roland's headquarters (north of Sanctuary). Approach the vault and take the ECHO Recorder from inside. Inside the vault you can also store other items which you don't need at the moment.
Leave Sanctuary and head to Three Horns Divide. Drive south, to the Frostburn Canyon passage. In order to get there, you will have to cross the bridge seen in the screenshot. The road to the teleport leads through a camp filled with Bandits, who you can eliminate from inside the vehicle.
Head onwards a bit and afterwards turn right. Stick to the left side of the canyon, as from there you will have a good position to shoot enemies in the below camps.
Head to the east of the canyon and afterwards enter the cave leading south.
After a few steps you should see an entrance to a cave branch on the left, which will take you to a chest with Class Mods. Take it and head back. Follow the cave south, clearing more Bandit camps and groups of Spiderants.
Follow the road pointed on the map. If you want to get additional experience points and items, you can of course check the other paths. However keep in mind that they don't connect with each other, as the map would suggest.
Keep heading deeper into the cave until you reach a fork in the road from the above screenshot . Take the left path.
Now you will have to look out for pipes coming out of the walls, ceiling and floor. Fire will start coming out of them when approached, so be sure to avoid it.
Cross the footbridge to the other side of the precipice and follow the tunnel to the cave with Firehawk . Approach her and hold down the use button (by default E) until the Reviving gauge fills up. Afterwards head north, take the Eridium nugget from the shelf and give it to Firehawk. You will have to team up and fight off incoming waves of Bandits.
After eliminating the last enemy, Firehawk will ask you for another Eridium nugget, this time found in the western part of the room. Give her the item to finish the mission. In return, you will receive a Class Mod. You will be teleported onto a path which you need to follow south-east, to a passage leading to Three Horn Divide .