In American English, "our way home" means "a method of getting home" - walking, a taxi, calling a friend. It's what you say when someone asks you something like "How are you getting home?" or "Do you need a ride home?"
"A way home" means the same thing.
"The way home" means the actual route - the roads and/or paths one has to traverse in order to get home. In a city, that could include taking public transport, but it would mean knowing which particular transport you have to take: a certain subway/metro or bus route/train: "The A-train" or "the number 11 bus" for instance. Simply taking "the subway" or "a bus" or "a taxi" is "a way home", not "the way home".
They are definitely not the same thing. Дорог is an apt translation for "the way" but not "our way".