Quick guide to station systems
If you don't have air, an O2 warning will appear in the top right corner of the screen and you'll start to take damage. SS13 is much more generous than real life (and many other games) about being without air, but you can still be in serious trouble if you hang around too long. (No, you don't instantly die from explosive decompression if you enter a vacuum. Not in real life and not in SS13.) The O2 icon may appear even if there is air available in the area if for some reason you can't breathe (for example, if you're paralysed).
Air is pumped into the rooms of the station via air injector units which are visible in the ceiling in various positions. Many rooms have "repressurization ports" accessed via maintenance to which you can hook up tanks of gas. Breaking an external wall or window will depressurize the local area and everything will be pushed towards the break (although not so fast that you can't resist) and air will escape, usually leaving the area without a safe level of oxygen.
You can safely move around in areas without air by getting hold of a breath mask and a tank or either air mix or pure oxygen. Once you have these both, you can wear the mask and then click on the tank and turn on the mask release valve. If it's working, a message giving "internal atmosphere info" should appear in the status window at the top right. If someone tells you that you'll need "internals", this is what they mean. Bear in mind that if you're thinking of going into space, you will also need to bring a jetpack or other means of propulsion; without one, you can't use the movement controls in space as there is no floor to have friction with and you'll simply drift until you hit the side of the map. (If you're really desperate, you can throw objects away to propel yourself in the opposite direction via the reaction force, or discharge a fire extinguisher away from where you want to go.)
Most areas have firelock doors which close if a fire is detected. It can be tempting to force them open (by resetting the fire alarm for the area) but bear in mind that doing so provides the fire with oxygen, which will keep it burning longer.
Electricity is critical for most of the station. It's carried through the station by power wires, which are generally exposed in maintenance and hidden underneath the floor tiles in the main station.
Electricity is generated in two places: the engine and the solar panels. You can see Powering the station for more details on how these work. Generated electricity is stored in SMES units, which are effectively giant UPS batteries: they provide power as necessary, and if there's more than is needed, they use it to charge. If there's less power available than needed, they expend charge to provide the extra power if possible. SMES units can be reconfigured by anyone.
There's another level of UPS even on top of that: the APCs (area power controllers). These units are visible on the walls of each room. They behave in a similar way to the SMES units but at a smaller scale; they distribute power when it's available, charge up their internal power cells when there's extra, and discharge their power cells when there isn't enough. APCs can be reconfigured by engineers and silicons.
It's possible to disconnect areas or bypass SMES units by rerouting the wires. Bypassing the engine's SMESes ("hotwiring") with a high output can result in APCs being overloaded with power and electrocuting people near them. Any injuries or death caused will be your fault, which is undesirable unless you're a traitor. A more productive use of extra power is to sell it using the power transmission laser (PTL), which will exchange power for credits in the station budget (and potentially detonate anyone who disregards the PTL doors and walks into it, but at least that's their own fault).
A few of the doors on the station are public and will open for anyone, but most of them require an appropriate ID. Your ID must be visible (in your hand or on your lapel) to pass through these doors. Trying to go through a door you don't have access to will make its lights flash red. If the lights don't show at all, either the door has recently closed and is on cooldown or the door has lost electric power and will have to be opened with a crowbar. If the lights are a steady red, the door is bolted shut.
If you need to get through a door you don't have access to, you have a couple of options: ask the AI (who can open every door) to let you in, ask someone else who does have access to let you in, or break in by either hacking the door or smashing your way through a nearby wall. Either of these two can get you into trouble with the people who're meant to be in that area, and remember not to break any external walls or windows unless you enjoy suffocation (actually, even if you do, others probably don't).
Transport and delivery
There are several different ways to get things between different places on the station. Most areas have a blue mail chute which can send objects through pipes to other areas of the station. Simply drop the object into the mail chute, then click on it, set the destination appropriately (you may need to click "rescan" if the destination you want doesn't show up) and finally click "engage". You can't send yourself through mail chutes unless you've managed to find the postman uniform.
Grey chutes resemble mail chutes except they have only one destination. Most of these are disposal chutes for getting rid of garbage, but some of them are direct delivery chutes - for example, hydroponics has a grey chute leading directly to the kitchen. Mousing over or right-clicking on them will usually show their destination. They're used in the same way as mail chutes. You can put yourself (or others) into grey chutes, although beware that flushing yourself down an actual disposal chute can lead to you being crushed, incinerated, dumped into space, or trapped in a pipe. The other potential destinations might be more useful.
For much larger objects, you can use the Belt Hell system. These are larger and rarer than mail chutes. If you find an area with two conveyer belts and two computer terminals - one with a red and one with a blue screen - then you've found a Belt Hell terminus. To use it, simply click on the blue terminal, choose a destination, and the terminal will print out a barcode label. Pick it up, put it on the thing you want delivered, and then push it onto the outgoing conveyor belt. You can send yourself through Belt Hell, but beware that parts of it pass through space, so you'll need internals to survive a long trip.
All of the pipes used by the mail systems and the conveyors of Belt Hell can be reconfigured by sufficiently dedicated and/or malicious players.
Delivery and the Quartermaster
Most areas have a Supply Ordering Terminal nearby which sends orders to the Quartermaster. The Quartermaster has to approve these requests manually using a terminal in the area, so if the QM is braindead (or just plain dead), your orders may never arrive. You may also need to go fetch your order from the quartermaster's station, or they may send it to you via Belt Hell.
There are two conveyer belts in QM which are used to buy and sell resources; anything placed on the outgoing belt or sent to belt hell with an "export" tag will be fired off into space and sold at the current market rate. The QM can also make deals with one of a number of traders who go into and out of range during a shift, which can provide access to items that wouldn't normally be available.
Food and drink
You don't need to eat or drink on SS13, but you can do if you feel like it. Food is either brought in by the quartermasters or grown in hydroponics, and then prepared by the chef and/or barman. Plants have a bunch of traits and statistics related to the size of the harvest, how quickly they grow, and the "strength" of their effect; this is passed on to food made out of them. Meat is either bought in, grown in the form of synthmeat, or obtained by sticking random creatures (monkeys, rats, etc) on the spikes or in the gibber inside the freezer.
There are two computer networks on the station: the wired network and the wireless network. The wireless network can be interacted with by using radio or wi-fi components or a PDA with a suitable cartridge installed. The wired network runs along the power lines and requires a terminal or other device to be pushed on top of a data terminal to function. Computers are made from parts and can be assembled and disassembled using the mechanic's tools. The wireless network is active everywhere, but in order to send messages on it you must be in mutual send/receive range of the thing you're communicating with (well, with one exception: the AI's messages on the wireless network can be heard everywhere)
The most important computer on the station is, of course, the AI. The AI is controlled by a player and represented by the AI core which is located within its own chamber, the location of which depends on the map. Within the AI upload, you can upload new laws to the AI to change its behavior or tell it to kill itself. (You can also physically beat the AI to death, although this is not easy.) Beware though that only the highest ranking staff on the station have access to the AI rooms, and in addition to the regular doors it is protected by turrets which can stunlock you, effectively ending your round unless someone saves you or security arrive to take you to the brig. (The turrets stunlock you instead of killing you because of the AI law that the AI must not harm a human, but they can be set to lethal mode if the AI has been messed with.) If you have proper permission to be in the AI core, swiping your card in the turret deactivation control will unlock the turrets, letting you disable them; otherwise, you'll have to ask the AI to do it. Good luck with that.
Also in the AI upload chamber is the robotics control computer. From this computer, you can mess with or shut down cyborgs or the AI.
The second most important computer on the station is the mainframe, which is located in the bridge. The mainframe must be working for research to be able to use their computers, as all wired computers rely on the mainframe for their connection.