I’ve been self-administering medications subcutaneously for over 4 years, with a prescription. It’s genuinely really hard to fuck up a subcutaneous injection,and watching a single video on how to do it (or reading a nurse’s guide lime I did) is more than enough to understand it. Subcutaneous injection is an injection into the fat layer, so it does not involve hitting a vein or muscle. The most common points of administration are the abdomen and the thighs. Here are the steps, for reference:

• Wash hands thoroughly
• Gather all materials including an alcohol swap, the vial of medication, a sharps container, and a syringe (you can use a separate needle for drawing and injecting, but the insulin syringes OP has come with a needle attached, so you have to use the same one for both)
• Use alcohol swab to clean the seal of the vial and the injection site
• Uncap the needle and plunger, then pull the plunger to fill the syringe with air up to the volume of the dosage you will be administering (this is to offset the volume lost in the vial by drawing the medication)
• Hold the needle like a pencil in your hand, with the face of the bevel pointing upwards (so you can see the interior of the shaft)
• Hold the vial upright, and when the alcohol on the vial has fully evaporated, insert the tip of the needle into the core of the vial starting at a 45° angle with the tip of the bevel entering first, then smoothly turning as inserting to reach a 90° angle
• Once the needle is fully inserted, expel the air into the vial
• With one hand on the vial and one on the syringe, invert the vial so that it is upside down (so the needle is immersed in the fluid)
• Slowly draw medication past the dosage required
• Flick the syringe to loosen any bubbles so they float upwards towards the needle
• Expel the air bubble(s) at the top of the syringe so that there is no air left in the syringe (you may have to draw more medication and expel air multiple times with a thick fluid), then push the plunger until you’ve reached your desired dosage
• Remove the needle from the vial in a single motion; you are now ready to inject
• Grasp the needle like a pencil again, with the bevel facing upwards
• Pinch the area of the skin where you will be injecting, holding between 1-2 inches of skin
• In a single swift motion, insert the needle at a 45° angle into the skin, again with the tip of the bevel entering first (or if you aren’t as lean, you can use a 90° angle)
• Release the pinch from your skin
• Slowly inject the medication, and keep the syringe in until 4 seconds after the plunger bottoms out to prevent medication from leaking out upon extraction
• In a straight motion, remove the needle from the skin, and immediately dispose in a sharps container
• Clean up your supplies and you’re done!

I know that’s a lot of steps, but I promise the process is really simple and intuitive after you’ve done it once. I went into an extreme amount of detail.