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An Old Friend
You will basically run right into this quest. When you leave the Detroit HQ for the first time, you should be stopped by Mrs. Reed. After exchanging some kind (or unkind) words, she'll ask if you are willing to look into the terrorist attack that killed Megan. I'm going to assume that you want to say yes. If you want a reward, you'll need to pull up the quest details branch and ask for it though.
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Cut to the Chase
You can start this off by talking to ex-detective Chase. I suggest that you do this first, since you'll be going to the police station for the main quest anyway. He's a security guard in a building right by the gas station. Just open the door, and then the security door to find him sitting at his desk. He'll be fairly cooperative. Make sure that you talk to him about what's happened and get all the leads that he has. You should know about a dirty cop, a bitter captain and a storage shed, along with a vital safe inside it, when you finish the conversation. If you have the persuasion module, you should be able to push him and get the code for the safe, if you want. It's not really that necessary though. You'll want to have your hacking skill at level three anyway.
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The Officers' Reports
There are three things you need to gather for the quest. The first two are in the police station. You'll presumably do this as you do the main quest. Just check in the series to find out the ways to easily get inside. I'll assume that you talked your way into the police station (just talk to Haas and use Absolve a lot), or that you're comfortable sneaking around the station with a bunch of cops wanting you dead. If you need help with that, check the page for the main quest.
The Dirty Cop: If you don't have the persuasion module, go up to the third floor and look for Chet's office, if you want to get the blackmail information (it's not necessary if you have the persuasion module). You can hack the door, but you'll need to watch out for the patrolling guard. You'll probably need to shoot him with a tranquilizer dart or stun gun if you want to do this in peace though. An easier way is to go into the office next to his, which should have a sleeping officer inside. Just walk past him and go through the vent behind him to get into Chet's office. You then just have to hack his terminal to get the email from the drug dealers. Just having the guy's name is enough.
Captain Penn: While you're in this area, you should also look for Captain Penn's office. You'll walk right past it as you head back down to the lobby to meet Chet. Just keep an eye out for the marker. It also has a vent leading into it, just look around the corner for the partially hidden vent in the wall. Just as before, look around for ebooks and then hack the computer to get some more information on the case and confirm the coverup. It's a level two hack, so you may need to upgrade. It's a good idea to do this as soon as possible. If you can't upgrade, you should be able to get in with the password "apophenion."
On your way out, talk to Chet. He's standing by the exit with a clipboard. If you don't want to use the persuasion option, just push him for information and you'll drop the blackmail on him. He'll then reveal his part in the coverup to keep you quiet.
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The Police Storage Shed
To rob the police storage unit, you just need to get to the back alley of the police station. You can get there in three different ways. Just go to the third floor of the police station and go out the vent to reach a fire escape that you can go down. You can also leap the fence right next to the police station with the help of a dumpster, if you have upgraded legs. You can also go through an electrified side alley. You can see it on the map, it's close to the sealed tunnel.
If you don't have the augmentation to stop yourself from being electrocuted, then grab a crate from outside and drop it into water. Grab the second crate that's inside the door and then use the two to make a makeshift bridge to get down the hallway, then jump to the dry spot and flip the breaker to make it safe.
The storage room isn't actually that well protected. You should have the code to get inside, so use it. You can then pick up the memos around the room to get more information about a lost witness, Dr. Reed's autopsy report and a note on the chemical used to torch the lab. The real treat is inside the safe though. It's a level three keypad, so wait until you have your hacking boosted enough (it should be a priority to reach three quickly, if you plan to explore much). If you really want to cheat, the code should be 7196. It's not vital, but it gives you a little more from the quest. Once the safe's open, you can grab Megan's bracelet and also find a research memo from her.
Once you have all of this information, you can go to your apartment complex to find Mrs. Reed. She'll be waiting in the lobby. Talk to her about what's happened, and choose your dialog as you wish. She'll thank you, and you'll net a lot of points from it.
- All information and screenshots from Deus Ex: Human Revolution