Report to Castle Dour in Solitude. Inside, you need to report to Legate Rikke. Talk with her and tell her you are interested in joining the legion. She will assign you to help take over Fort Hraggstad.
Clear out Fort Hraggstad
With the assignment given, it is time to head out to the Fort and start clearing it out.
One of them will try to warn you off. Simply keep on the approach to rush in. You will have plenty of them to fight through. You will be dealing with Bandit Marauders, Highwaymen and Bandits. There could be other variants because they are randomly generated.
Clear each section of the fort carefully. There are only a few bandits in the courtyard, only one or two just outside the entrance and another couple on the walls and parapets.
Clearing the walls and courtyard just leaves any at the top of the tower. Head to the Western wall of the Fort and clear any Bandits from there. After that, you will need to make your way up the tower and check the top. There is likely one Archer up there you will need to defeat.
With all the bandits on the outside clear, it is time to head inside. Drop down to the courtyard and look in the Western center portion. You will see the door leading inside. Just inside you will find a bandit standing in front of a hearth. Deal with them and head to through the Northern door.
There are more Bandits in this area to the East that will come after you depending on how noisy you are about the first bandit.
One of them is the Bandit Chief, the leader of the group. They are going to be your final target. Eliminate them and then clear out any remaining Bandits.
Head outside and turn to the Northeast. You will find a second indoor section. There are more Bandits to be dealt with in there.
Inside here you will find more bandits. There are 2 in the first room that you need to deal with. Further in is a third Bandit that you will need to deal with. Be aware of them and listen for their approach. When all the bandits are dead, the objective will update.
Report to Legate Rikke
Take the oath
Talk with General Tullius to take the Imperial Oath and be sworn into it. Just go through the various responses to be sworn in.