Crafting syringes requires plants. These are scattered all over the islands, and are impossible to miss. The best and easiest ones to craft are the Medicine as they only require 1 Green plant. If you have anything else to hand that can be crafted into any other type of syringe, then you might as well use it.
Overall, this achievement will come naturally as you progress through the game. A good boosting tactic is to stock up on Green plants and find a predator animal. Let it attack you, then press and hold to heal yourself. Keep doing this and when you run out, go craft some more and then repeat the process.