A new camera, or even and old one that you haven't used that much, can be a bit daunting. Menus and lists full of options with unhelpful names and a manual that's either too thin or too thick or just too boring to be useful.
These are the key functions that I'd find first (straight after turning off that annoying bleep that accompanies every button press):
Hopefully you're already familiar with the mode dial. It's the A/Av (Aperture Priority), S/Tv (Shutter Priority) and M (Manual) modes that are the interesting ones. Get used to choosing your own settings rather than letting your camera decide. If you're unsure, start with Aperture Priority and get used to how the different apertures change the amount of the image that's in focus (there's a simple explanation on YouTube here).
Note: Most Fujifilm and Leica cameras don't include a mode dial, just the choice to set either aperture or shutter speed into 'auto' mode.
In shutter or aperture priority modes, your camera is trying to make your photo 'averagely bright' across the image. Often that's fine, but sometimes you want a brighter or darker image. Using Manual mode is one option, but you can also use Exposure Compensation to tell the camera that you want the image to be brighter or darker than average.
Most of the time you'll probably just want to take one photo at a time (Single shot mode). If you want to take more quickly (such as for fast sports) then your camera should have 'Continuous' mode(s) that allow you to keep your finger on the shutter and take multiple shots. It will also have 'Bracketing' modes, where you can use slightly different settings for each of 3 (or 5) shots taken in quick succession.
You can generally choose whether the camera focuses on a single point in the frame (and move that point around the frame), or a wider zone. A smaller focus point gives you more control and accuracy, but the focusing may be slower as the camera has less information to work with. Zone focus modes allow the camera to analyze a larger area of the screen and identify objects to focus on.
Single AF means that you lock the focus by pressing the shutter halfway. Once the camera has focused, you can then reframe the shot whilst keeping the same focus distance. Continuous AF means that the camera will be constantly refocusing while you hold the shutter halfway, so can be better for moving subjects. I mostly use Single AF, even for candid street photography.
Sometimes your camera might not focus where you want it to. In low light for example, or if you're shooting through a window or through the leaves of a tree. It's good to practice focusing manually just in case you need it, and your camera might have functions to help you get it right, such as a magnified display of the focus area, or 'focus peaking' (in-focus items are given a coloured highlight in the display). It can also simply be an enjoyable experience to take back control from the camera and do everything yourself.
RAW and JPEG settings
This is the format of the files that the camera is recording. RAW files contain the most information if you're going to edit the images later but are also the largest files and have to be 'converted' by software to create a final image. JPEG files are smaller but give you less flexibility if you want to do much post-processing in software like Photoshop. JPEGs typically have different size settings too so make sure your camera is set to the highest quality settings.
Formatting the card
Once you have your photos stored in at least two other places (highly recommended!), you'll probably want to delete them from your camera. Re-formatting the card is the cleanest way to completely remove everything from the card and is worth doing occasionally, if not every time you want to delete everything from it.
Not just for those family selfies, self-timer is a great tool if you're using a tripod but don't have a remote shutter release. Setting a short timer takes away any vibration that you otherwise cause simply by pressing the shutter button.
Many cameras have automatic ISO functions where you can also set a maximum ISO value and a minimum shutter speed. This lets the camera choose an ISO value in order to maintain the shutter speed that you've set.
Reasonable ISO range for the camera
This is one you'll need to experiment with because each camera is different when it comes the quality of the image at higher ISO settings. For example, my Fujifilm is fine up to 1600 and I'm often happy with 3200. Beyond that there is generally too much noise in the image. Experiment first so you know what ISO limit you're happy using. Regardless of the camera though, image quality will always be higher at lower ISO values.
If you're new to photography I hope that's not too daunting. One of the best things you can do is just to experiment. Try a few shots and you'll soon get the idea of how each functions works. There are also some great overview videos on YouTube for individual camera models that can make it easier to find the relevant functions than searching through your manual.