I use Gyazo for images and Fraps (full version) to record videos because I can easily record Minecraft videos at 60fps and they're good quality.
Is it just me or is my Fraps recording at low quality, because the bitrate is very high but the video is blurry. I use Mirilis Action now and the quality is way better
For high quality videos make sure you're recording in full-size. The videos will appear much higher quality and still stabilize the recording at 60fps. If you're recording in half-size, which I assume you are, the video does look low quality and blurry.
As has already been mentioned: For websites I usually use the snipping tool, or the regular print screen function. For in-game screenshots I just use F2 (or Fn F2). I know this has already been mentioned, but if you're looking for functions that are regularly used by many players, another opinion doesn't hurt. c:
I am not recording in half-size. I have make sure that I'm recording in full-size many times, and the quality is still poor. :/ If you take a look at my videos, you will see what I mean.
Ah. Screenshotting: When you want to take a screenshot, simply press the f2 button located on your keyboard. If you want to upload the image you can use a site like gyazo.com or imgur. Recording: I suggest bandicam, or fraps.