Gone are the days to use third-party screen capture tools and save your screen in Quicktime. Apple includes built-in tools for taking screenshots and recording videos in macOS Mojave, and they are very good.
Keyboard shortcuts to know
Although you can technically launch the screenshot tool from the "Other" folder of the Launchpad, it is best to learn keyboard shortcuts. You can change all these combinations in the Shortcuts pane of the keyboard preferences, but these are the default values:
Command + Shift + 3: Saves the entire screen to a file on your desktop (and will also display it in the lower right corner of the mouse to drag it into different applications). You can hold down the Ctrl key while doing this to save it to the clipboard only, so that your desktop remains clear.
Command + Shift + 4: Opens a selection menu in which you can draw a frame around what you want to select. It will also be saved on the desktop and you can also hold Ctrl to copy only to the clipboard.
Command + Shift + 5: This combo is a special case. It opens the main screen capture options bar where you can access all the settings and different tools:
From left to right, the tools of this bar:
Capture the entire screen.
Capture a specific window and automatically crop the image.
Capture a selected part, and is the default option.
Start a recording of the entire screen.
Start a recording of the selected part of the screen.
The options menu also includes additional settings that allow you to choose items such as where the screen capture is saved and the use of a timer.
Once you are done, you can press "Capture" or "Save", or simply press Back. Do not forget that if you are recording a video, you have to reopen it to stop recording or press the stop button on the menu bar.
If you want to trim or edit the video, click it in the lower right corner of the window to open it in Quick Look, and then click the ellipsis button.
This will save the video in place for you to send it.