Firstly, to answer your question. "cachi" is a sort of persimmon, a type of fruit that looks like a plump orange tomato. There are two types, one better known as "persimmon" in english, harder, less moist one, there is another type, extremely juicy, sweet and soft, that melt in your mouth. The latter is absolutely delicious!! I have never seen a jarred version, so I am not sure how they would taste... what are they immersed in, syrup? oil? or something else?
"mosto cotto" is a sort of confection made from grapes, to be more exact what is left from making wine. It has a consistency of thick preservative, not too sweet, however there is no comparison to those ordinary "grape jellys"!! It has a distinctive rich flavour, my favourite way to use it is to top fried plantains or sweet potatoes with it. Also nice on pancakes, blini, waffles, crepes, etc. coupled with honey or whipped cream.
Yeah, I had the similar experience when I lived in Spain for 3 years. I was mostly inside the US naval base, and most Spaniards I came in contact spoke English, so I never got a chance to learn much of the language, and I sorely regret it...
You know, there are a couple of other members here who live in Milano, RDG is a native Milanese, and Lulu, an English girl just moved to Milano with her Italian husband. So maybe they can help too if you have more questions about either city or Italian products. I love it in Rome, and I am glad you enjoyed your stay here!!