Briefly, metal nanoparticles are generally synthesized by reduction method using sodium borohydride, as in the case of silver and copper. Gold is mostly synthesized with the classic Turkevich method. Some nanoparticles can be synthesized using ethylene glycol (ruthenium, for example). If we are for metal oxide nanoparticles (SiO2, Fe2O3, CuO, etc ...) can be used precipitation method or sol-gel. The issue of stabilization, polymers are often used (PVP). Sodium citrate and other organic substances as well. That takes away a lot of the route and reagent availability.
Silica nanoparticles may have their surface modified to have stability or compatibility in some specific medium. This depends a lot on the application.
You will have to research thoroughly to understand each method well, as they have particularities and limitations. There are variations, some use sulfates, other chlorides, organosilanes, what matters is the precursor ion of the element.
I hope this answer will serve more as a guide to the various methods.