Overview
All the information you need about shortcuts, what you need to unlock them, how to use them, and some other spoilery things.
How to unlock shortcuts
The first step to unlocking shortcuts is to play the game like normal until the Sherpa (blue man with a round head) shows up in the Gungeon. After talking to him in the Gungeon you need to go to the Breach and talk to him again in the lower right area. He will now show up in the elevator shafts left by the bullet elevators at the start of each chamber. He starts on chamber two, you will be able to tell which chamber he is in by a yellow sign on the wall above the hole.
You will now need to bring him the items detailed below in any order, although he will ask for them in the order listed. Each individual requirement must be met completely at one time, no bringing him half of a requirement one run and the other half on another. Multiple requirements can be met in a single run.
After bringing him everything he needs on a chamber a shortcut to that chamber will be unlocked in the room in the lower right part of the Breach. If you talk to him after unlocking the shortcut you will unlock a new item and he will move on to the next chambers elevator shaft where you can repeat the process with more demanding requirements.
Chamber 2
Requirement 1: 3 Blanks (Found in many places, start each floor with two)
Requirement 2: 3 Keys & 120 Casings (More info below)
Requirement 3: 10 Hegemony Credits (Earned by killing bosses)
Requirement 4: Master Round, First Chamber (Kill the first boss without taking damage)
Unlock: Gunboots (Shoot bullets backwards when you dodge)
Chamber 3
Requirement 1: 3 Armor (Found in many places, Marine starts with one.)
Requirement 2: 4 Keys & 180 Casings (More info below)
Requirement 3: 15 Hegemony Credits (Earned by killing bosses)
Requirement 4: Master Round, Second Chamber (Kill the second boss without taking damage)
Unlock: R2G2 (Follows you around and shoots at bulletkin)
Chamber 4
Requirement 1: 4 Junk (Chance to drop after breaking a chest)
Requirement 2: 5 Keys & 240 Casings (More info below)
Requirement 3: 20 Hegemony Credits (Earned by killing bosses)
Requirement 4: Master Round, Third Chamber (Kill the third boss without taking damage)
Unlock: Gungine (Automatic gun that increases move speed when held)
Chamber 5 & Boss Rush
Requirement 1: 6 Full Heart Containers, does not take them away (Kill bosses without getting hit)
Requirement 2: 6 Keys & 300 Casings (More info below)
Requirement 3: 25 Hegemony Credits (Earned by killing bosses)
Requirement 4: Master Round, Fourth Chamber (Kill the fourth boss without taking damage)
Unlock: AKEY-47 (An AK-47 that fires keys capable of unlocking things like regular keys)
How to earn more keys & casings
The three most important things. Never use keys and only spend casings on keys if you really have to. Play pilot so you can buy cheaper keys and get items without using keys. Finally go to both secret chambers for more drops.
The first secret chamber can be accessed on chamber one by rolling a water barrel into a fire place which opens a secret room with a trap door in it. You will need to open two locks to open the trap door, the lockpick can be used to save keys but if it jams you will need to restart the run. The second secret chamber requires you to bring a piece of special armor from the first secret chamber to an alter on the second chamber. If you get hit while holding the armor it will break so you should wait until you complete the first secret chamber before picking it up.
It’s also useful to sell any items you do not need to survive by dropping them in to the floor grate that can spawn in shops. Passive items can be dropped by clicking on them on your map screen then clicking drop in the top left. Junk and master rounds can be sold.
Other than that make sure to look for secret rooms and avoid being hit since enemies that have hit you will not drop casings.
Other Info
You can access the shortcuts you have unlocked by going to the room in the bottom right of the Breach and using the buttons on the floor to select the chamber then walking in to the elevator.
If you have unlocked the chamber five shortcut you will also have access to an elevator that takes you to boss rush. You must pay the Sherpa 3 Hegemony Credits each time you wish to use the boss rush elevator, the first time is free.
If you take the shortcut to chamber three or higher a very generous “man” will have a selection of three weapons for you. On chamber three you can take one of the weapons, on chamber four you get two, and on chamber 5 you get to take all three.
Thank you to all the people in the comments that have corrected problems with the guide or asked questions about things I forgot to mention in the guide.