Government District Overview - Baldur's Gate 2
The Government District is one of the more uneventful districts that you will visit in Athkatla, mostly all you're going to find here is dialogue and no combat. You'll meet 2 companions for the first time in this district, Viconia and Jan. Both of them will need your help escaping trouble when you first encounter them, so long as you help them they'll be available for recruitment whenever you want after that.
Most of the buildings here are involved in quests and can be ignored until you're on a quest that sends you to them. Almost all of the buildings are marked too when you open your map making finding them quite easy when the time comes. One of the more important things to do in this district is to speak with Delon or with Tolgerias in the Council of Six building. Either of these NPCs will unlock Umar Hills on your World Map which is an important location.
Viconia: When you pass by this location for the first time there will be an event with a bunch of angry protestors trying to burn Viconia at the stake. You'll want to either attack the fanatics and then click on the stake to free Viconia or just click on the stake. Either way, you'll end up fighting the fanatics to free her. From this point on Viconia will be available as a potential companion.
Delon: Speaking with Delon will start the Investigate the Deaths in Umar Hills side quest. This quest is the main chain involving Umar Hills and is also one of the two quests that can unlock Umar Hills as a World Map location for you. I strongly recommend you pick this up as you pass through.
Jan: Jan Jansen is a potential companion in BG2. Words are not enough to describe how great this companion can be with his humorous dialogue. The first time you pass through this area Jan will stop you and start a scene during which you'll be approached by Trax. You have a few different outcomes depending on what dialogue options you choose, if you accidentally send Jan to jail you can pay his bail at the Jail in the Government District. Alternatively, tell Trax you were just talking about the weather and Jan will become available as a companion from then forth.
Council of Six: There are a few named NPCs inside of this building that are involved in different quests. Most important is probably Corneil whom you can buy a magical license from which will allow you to cast spells in Athkatla. I also recommend talking to Tolgerias to start The Hunt for Valygar Corthala side quest as that will mark Umar Hills on your World Map if you do not already have it.
Firecam Estate: Home of Keldorn and his family, you'll be visiting this house during Keldorn's Companion Quest.
Delryn Estate: Home of Anomen's father and involved in Anomen's Companion Quest.
Deril Estate: The Deril Estate is involved in Cernd's Companion Quest, you'll be visiting this location to meet/acquire Cernd's child.