The Manor Cliffs portion of Deathloop will be your introduction to a lot of the systems in the game. As you make your way through, you will learn all manner of useful bits of information about the game. From how to use weapons, to how to crouch, all the basics are covered before you reach the Manor Cliffs themselves.
Once you get out on the cliffs, you will come to a large building ahead, with some guards between you and it. Place your reticle on the guards and hit the down arrow to track them. Drop down a level to the road below, then hack the radio on the vehicle to the left. This will cause a nearby guard to check it out, and you can take them out.
After that, simple timing will get you the rest of the way with the other guards. If you are spotted, you can rush them with your machete, or take them out with your gun. After that, you need to get into the library, which is quite easy to do.
Head for the front door and taking out the Hackamajig you got earlier, then hit the R1 button to hack the sensor above the door. This will allow the door to open, and you can head inside. Move to the back left of the building and climb the stairs to the top. Pull the red level on the path left to progress the story, and we will avoid dropping any spoilers here for you.