This guide is meant to help you take better looking screenshots and avoid issues which, at least in my experience, show up fairly often.
Not every screenshot needs you to go through all these steps. Obviously there’s a difference between a quick snapshot of a work in progress to share with your friends and a screenshot you put onto a community content upload to really show off your build.
These tips are written first and foremost for screenshots of stationary ships meant to show off the creative side of the game.

All keybinds mentioned are default keybinds at the point of writing.
All builds shown in example screenshots without credits are entirely my own.
This is an updated and improved version of a guide I posted on the game forums in 2019, special thanks to Plaz and Lancia for providing some builds and helping with the draft.

These tips apply to the vast majority of screenshots taken.

1. Hiding the UI. This is pretty easy. Just press F1 + G before taking a screenshot. While the game provides a keybind (F6) for taking a screenshot without UI, this doesn’t hide all UI elements, most prominent are symmetry planes and the yellow build mode box. Additionally, this method of hiding the UI supports all options of taking screenshots. Once done, press F1 + G again to bring back the UI.
   
   The same screenshot taken with F6 without (left) and with (right) hiding the UI with F1 + G
   
   
2. Basic Positioning. Make sure your build is positioned in a way that it is easily visible against the skybox and any background objects

These tips aren’t necessary for quick snapshots, though I consider them necessary for any screenshots taken for the purpose of properly showing off a finished build.

1. Lowering the FOV. This helps your build appear less distorted towards the screen border. It is easily accessible, just press Esc, go to Options, then go to the “Graphics” tab. There you’ll find “Field of View” most likely with a default value of 85. Lower this to 45 or less, then apply. I strongly recommend returning it to 85 once you’re done taking screenshots.
   Additionally, you might have to use a higher value for large builds or screenshots containing many LOD blocks (e.g. pipes) because at the same distance lower FOV screenshots appear zoomed in. This can result in either the LOD distance kicking in before you have everything in frame or - in the case of very large builds - distance related artifacts to appear.
   A side effect of this method is that a smaller area of the background is visible.
   Lastly, I recommend to avoid this for most interior screenshots, you tend to run into a wall before you get everything in frame.
   The top screenshot was taken with an FOV of 85, the bottom one was taken with an FOV of 45. I kept the position the same, though I did zoom out (L. Shift + scroll wheel) so the entire ship is in frame.
   Build by Plaz and Lancia
   
   
2. Advanced Positioning. Finding a system with a background that fits the mood you want to go for. Besides the background, you might want to consider whether you want a star or a void system. Stars can be nice, though they can also result in bad looking lense flare.
   Once you have done that, you’ll want to find a sector in that system with a lighting direction you like or rotate your ship until you have the light coming from the desired direction. Depending on the mood you want to create or what you want to show off (e.g. details utilising glow) you might want to take a screenshot from the dark side of the ship.
   The positioning allows the heavy utilisation of glowing elements to shine. The system background fits the build and its excessive edginess.
   Build by Plaz, Lancia and me
   
   
3. Basic scenery setting. This is fairly simple. Just make sure you don’t have any entities in frame, that aren’t meant to be there (e.g. a stick shop)

Most of the time, using build mode is the obvious choice for taking screenshots. However, there are some situations for which this doesn't hold true. Here are a few alternative methods which are situationally preferably to build mode.

1. Astronaut Mode. For parts on an astronaut scale (most prominent example being interiors) it can help to take screenshots in astronaut mode to match the experience of somebody interacting with the build. An additional plus is that when combined with Free Cam (see next tip) you now have an Astronaut to include in your screenshot.
   Dave interacting with some consoles
   
   
2. Free Cam. While in astronaut or flight mode, you can double tap R. Shift to rotate your cam independent of your astronaut/ship. This is especially useful when zooming out (L. Shift + Scroll Wheel), allowing you to take screenshots of an astronaut interacting with the interior or things like thrust plumes without having to set up a camera specifically for that.
   Drawbacks are that this can behave a little odd depending on ship alignment and that you are just rotating around your current camera (Astronaut, Ship Core, Camera block).
   Using Free Cam to show thrust plumes from a more interesting angle than straight behind.

These require more effort than previous tips and are certainly not necessary by any means. Consider these nice things you can do if you feel like spending some time on taking screenshots.

1. Using a dedicated camera ship. This doesn’t have to be a complicated build, technically just a core will do (though I’d recommend at least going for the simple 3 block stick). Flying this thing around your build allows you to get additional angles not possible with build mode etc. due to having access to rotation controls.
   While perhaps not immediately noticeable, this shot is only possible thanks to being at a different angle than the ship.
   
   
2. Advanced scenery setting. There’s basically no limit to your creativity when it comes to this. Once you’ve made sure there’s no unintended objects in frame (see “basic scenery setting”), how about adding some intended objects? Maybe you want to show something in the process of docking to your station’s docking tube? Maybe your fighter would look cooler with some friends?