You start in the first part of the underground [South Quarter / Riverside - Warrens] and you need to keep precautions because there are two guards around. It is some sort of facilitation that the enemies should not leave their posts, by default and so, you do not need to keep an eye on them, on a regular basis. Of course, I recommend that you take a look around the first location in the underground, because you can find here a Wine Glass - 3G and a Brooch - 7G, which has been hidden in the tunnel behind the back of the opponent with a torch.
For the first time here, you need to decide which way you want to cross the underground. A very good idea is to take the door on the left. It is barricaded by default, but you can climb to reach the crate shown in the above screenshot. This requires you to use the Wirecutter here and, for sabotaging the device, you will be rewarded with deactivation of the blockade on the door below. Explore the side room and find a Broadhead Arrow, a Fire Arrow and a Poppy inside.
Remain in the small room and locate the opening near the ceiling. It is a good idea to climb here, thanks to which Garrett will reach a side passage. Take a look around the area, because there is a collectible here (The Pinned Castinets - Jewelwing). Take to the exploration of the side path, thanks to which you will walk around guards, who are watching the main corridors, and you will reach a big vertical shaft.
You may have guessed that you will require a rope arrow here, which you need to aim at the interactive point in the ceiling. Jump to reach the rope, but you need to limb first, to reach to the top ledge shown in the above screenshot. Here, you will find a valuable Ancient Vessel - 48G. Only then, climb down the rope, to the very bottom, thanks to which you will end up next to Passage B, which has been described further in the text.
If you do not have the wirecutter on you, or if you do not want to take the side-passage described above, you can remain in the starting point. Carefully, walk around the two guards and reach the exit. Follow the main corridors and collect the Wine Glass - 3G. The main path will take you to an area guarded by two more guards, and additionally, you will have to avoid the shattered glass. Note that only one of the enemies is patrolling this area. His colleague (the one with the torch) stays put in one position. I recommend that you distract this guard by throwing an item, and also consider robbing him off two Coin Purses - 5+7G. In general, however, the most important thing is that you wait for the right moment to reach the stairs downwards. The stairs will take you into the region of Passage B which was described further in the text.
The last possible variant is, in my opinion, the most interesting. It will allow you to walk around the enemies in the second group and start the exploration of the second part of the underground, in the best possible spot. Also here, you need to walk around the first group of guards carefully and follow the main corridors. Stop in the position shown in the above screenshot and use the wrench to open the venting shaft. Taking this route will allow you to reach the horizontal wooden beams above the main path. Apart from that, you will soon take a turn to the East and reach a side room, which is unattended by anyone. Turn North now and go towards Passage A, which was described further in the text.
To sum up, taking passage A is possible only if you have opened the grate of the venting shaft and walked over the wooden beams. You now are in the eastern part of the underground and, after you slide down, you will start the exploration of the second part of the underground, in a better spot.
If you have taken passage B, it is possible that you have taken one of the two remaining paths. I recommend against following the main corridors, because you then would have to avoid enemies unnecessarily, and you would not find any loot. It is a much better idea to gain access to the side passage, which will gain you additional loot and let you walk around all of the guards. Both of the paths will take you to the second part of the underground, where you will start the exploration a bit further away from your destination.