The first two Soviet movies that I ever bought on DVD (because I liked them so much on grainy old VHS tapes!) were 1973's Иван Василевич Меняет Профессию ("Ivan Vasilyevich Changes Professions") and 1967's Вий ("Viy" -- the name of a demonic monster who shows up at the end of the movie).

I definitely recommend both of these to learners because the DVD versions available through Amazon have excellent-quality subtitling in both English and Russian, so you can watch it first with the English subs to familiarize yourself with the characters and plot, then again with Russian subs for read-along practice with the dialogue. (Finding Russian movies with Russian subtitles is, alas, not always easy, so any DVD that has this feature is worth seeking out!)

Plus, they're both fun movies -- a bit campy and dated, but in a good way. "Ivan Vasilyevich" is a time-travel comedy that's set partly in 1970s Moscow (i.e., the Brezhnev era) and partly in 16th-century Moscow, and "Viy" is a wicked-witch thriller set in (I think) 18th-century Ukraine. "Viy" is also somewhat famous as the first [and only?] "horror" movie produced under Soviet censorship. (Though it's FAR from scary -- I doubt any kid older than 10 would get nightmares from it -- especially when you consider the ketchup-spattered gore films that were beginning to come out in the US, UK, and Italy in that era.)

P.S. I'm not at home right now, but I think that both of the DVD versions that I recommend (i.e., the ones with the good subtitles, and also very good video/audio quality) are distributed by the "RUSCICO" label -- you might find these films through other companies because they're so old, but I can't vouch for other versions.

P.P.S. If you want a WWII movie, Летят Журавли ("The Cranes are Flying") is definitely one of the all-time classics. Unfortunately, I don't have that one on DVD yet so I can't recommend a specific version to look for.