Docking at and Launching From Stations - Don't Get Yourself Killed, Rookie

A fast way to get yourself killed in this game is to approach ANY station and enter it without permission. If you enter a station without a docking request, and then get scanned, you will be terminated quickly by station defenses. Bye bye. See ya later. Don't let the mail slot slap your remains on the ass as the Automechs scoop you out of the station.

Silent Running
If you want to enter a station without permission for some reason, doing so requires silent running to avoid being scanned. Details can be found on the wiki here:

Wikia Silent Running

How to Request Docking
As long as you have sufficiently good reputation with a station's controlling faction, you should be able to dock there. You just have to ask nicely. 

• Get within 7.5 kilometers of the station
• Open your left hand HUD, go to the Contacts tab, select the station, and then select "Request Docking"

At that point one of two things should happen:

1. Docking request denied: Your request can be denied for various reasons:

• If you are outside 7.5 kilometers from the station
• If all docking bays are currently occupied, you will not be cleared to dock; try again when you see a ship leave, or change your game instance to another mode (open, solo, private)
• If the station does not have a landing pad big enough to fit your ship
• If you do not have sufficient reputation with the station's controlling faction (they are "hostile" toward you)

2. Docking request granted: If there is an available landing pad that fits your ship's size, you will be cleared to dock and a ten minute timer starts to count down. When the timer expires, so does your docking clearance.

The docking bay number is displayed above the timer. Get yourself through the mail slot and get your belly on the bulls eye of your assigned dock.

My docking tips are here:

The Humble Docking Computer and Manual Docking

How to Launch
Once you are nestled safely in a landing bay, and have finished your station business, there are two options on your HUD below Starport Services:

• Return to Surface
• Launch

Return to Surface: This option simply returns your ship to the surface of the docking bay where you can see the inside of the station (or the space surrounding the platform). It does NOT launch your ship.

Launch: The launch option can be selected from either inside or outside your docking bay, and it will release your ship from the docking clamps. A five minute timer starts to count down. You don't want to still be in the station when the timer reaches zero. Docking control operators are dicks, and they will activate station defenses and destroy your ship if it is caught loitering in their air space. 

Loitering at a Station too Long
If you decide to ignore the five minute countdown, here are the last six minutes and thirty-five seconds of your life as a pilot. The part of our ill-fated ship will be played by the Sidewinder I bought just for this purpose. We will name her Lucky Girl, and by the magic of my mind, we can hear her thoughts as you fumble around and get her murdered.

Lucky Girl during a happier time in her short life:

"This is going to be so AWESOME! My first pilot assignment. 'Don't you want to be an Imperial Clipper?', they asked me at the factory. Hell no! All flash and no grit. Put my AI brain into a Sidewinder. The backbone of our galaxy. Put me in the trenches where I can help a new pilot learn to fly free. To be one with the galaxy and balance their chi. Undocking...here we go! I'm so excited I think I just flushed a little fuel."

LEAVE STATION - 4:49

"That's right. Ease me up, little buddy. You are so cute in your flight suit. I think I will call you Smoochie. Oh! I'm so proud. You remembered to retract my landing gear! Okay, now let's head for the mail slot. That's it...no...where are you going? The mail slot is the other direction. Okay, you just want to look around. I get it. ITO Orbital was converted by the Federation forty years ago into the agricultural station you see all around you. They supply enough food for five systems."

LEAVE STATION - 3:07

"Wait...no...Smoochie, we can't loiter above another landing pad like this. They are going to issue you a..."

"...yeah. A loitering warning. You have thirty seconds to move us away before the station opens fire. Okay, whew! You had me going on that one, Smoochie! I love how we kid each other like this. We are so muy simpatico! Now you might want to move us toward the landing slot so we can exit the station. That timer is getting lower. Oh, that? Yes, you just got a fine for blocking the landing pad. Chump change for a great pilot like you. Shrug it off and let's get a move on! Lots of galaxy to see outside there! Come on, buddy, let's go!"

LEAVE STATION - 0:43

"Smoochie, my com padre. I don't mean to be pushy. We only have forty seconds until you get fined for trespassing. Yeah, okay, I can see how a pilot might want to try and fly down into the agriculture holding pens like this, but there is a shield which keeps my hull from being able to...ow! Shit! Okay, that hurt a little bit, even with my shields on. Just ease me back...there you go. Um...we should really be..."

TRESPASS WARNING - 1:30

"Okay, see...there you go. Now I can see how a new pilot might be wanting to get the full experience of every possible station warning available to you. Run the gamut, so to speak. I would have done air quotes on that last one, if I had hands. But that warning you see right there is the last one we see before we die in a blaze of laser fire. "

TRESPASS WARNING - 0:21

"Hey, dick weed! They don't fuck around with that warning. Why are we just sitting here? It's like you have a death wish or something. Wait, do you have a death wish? They didn't teach us counseling techniques at the factory, but if I had to make an educated guess, I would say YOU ARE BAT SHIT CRAZY AND TAKING ME WITH YOU! Does this make you feel like a man? Killing an innocent ship for some perverted..."

Full disclosure: A sidewinder was seriously harmed during the photo shoot for this article.