"Sin" is "without", yes; think of it as the opposite of "con". Sino is "but", but it is used only in a specific type of context which I will further describe; "pero" is "but" used for every other type of context.
Sino is used to demonstrate "Not one, but the other" types of sentences. I'll give some examples: "No quiero vino, sino agua" = "I do not want wine, but water", meaning "I do not want wine, but I want water" or "I do not want wine, but water instead." "Yo no prefiero beber, sino comer" = "I do not prefer to drink, but to eat", meaning "I do not prefer to drink, but I prefer eating over drinking."
On the other hand, "pero" is used for every other context of "but" as far as I know "Me gusta comer, pero no quiero ahora mismo" = "I like to eat, but I do not want to right now." "Quiero un perro, pero yo tengo dos gatos" = "I want a dog, but I have two cats."
Hope this helps :)