Learn Italian: Cerco un cappotto usato da mettere in inverno. - I am looking for a used coat to wear in winter.

When an infinitive verb describes the purpose or function of a noun right before it, Italian uses the preposition da. Here, un cappotto da mettere means a coat for wearing. You will also see this pattern in phrases like qualcosa da mangiare (something to eat).

Does mettere mean to wear? I thought it meant to put.

You are right that mettere literally means to put. However, it is very commonly used in everyday Italian to mean to put on or to wear clothing. You could also use the specific verb for to wear, making it un cappotto da indossare, and the meaning would be exactly the same.

Why isn't there a word for for after Cerco?

The Italian verb cercare already means to look for. You do not need to add a preposition like per. You just connect the verb directly to the object you want, like cerco un cappotto. Adding per is a very common trap for English speakers!

Why is it in inverno? Do all seasons use the preposition in?