@postman @swee'pea99 and
@Supersuperleeds have already recommended it, but I'll add another vote for
It's a Wonderful Life.
We're approaching the time of year when our copy will be dusted off and put on again for it's annual showing. a truly special film.
Another excellent Jimmy Stewart film (and in B&W) is
Harvey which I can watch time and time again.
Pretty much any of the Ealing comedies -
Whisky Galore, Passport to Pimlico, The Maggie, etc, etc.
Try
Hell Drivers for something a bit less humorous (although it does feature Sid James, and a young Sean Connery) or
Dambusters
If you watch the second channel, BBC2 often has a couple of old B&W films on early on Saturday and Sunday mornings too and Talking Movies has also been mentioned already - there is some rubbish on there, but there are gems too.