How Do I Make Schnapps?

Well, you can infuse almost anything. Especially in vodka. The trick is to find ingredients that take to it. It might be difficult to produce these flavors effectively because fresh peaches could get icky and the mint's tannins could become very bitter. You could try spearmint, which might be a little less harsh when infusing, and dried peaches will reconstitute somewhat and lend flavors. Russians do a lot of infusing with vodka, most successfully with things like ginger, cucumbers, berries and my favorite horseradish (don't knock it till you try it). You might also have to add things like sweeteners (agave syrup is a good one, or simple syrup) and spices depending on what you're doing. Keep in mind, Schnapps is technically brandy, meaning, it's a sweetened, flavored grape-based spirit. What you're making here is infused vodka, not the same thing. 

Start with a decent, flavorless one, like Stoli, Absolut or Russian Standard and go from there. 

You could do this with rum, gin, tequila or brandy too, keeping in mind the base of the spirit will always retain its own flavor. What you put in it needs to match it to begin with. Mint rum? Ew. Strawberry tequila? Si, por favor.