Strathaven for any particular reason? The Clyde Walkway would take you to Lanark, or you could drop off it somewhere around Hamilton and head south (along the Avon to Stonehouse?)
I mean, as far as I can tell from Google maps, all those roads have pavements. And if you want to keep west and pass through Glasgow, rather than heading over to the Pennine Way, you should - it's a great place. But unless you absolutely have to take the shortest route, I don't think I'd go straight through East Kilbride, and those look like pretty dull roads. Could you even skirt EK to the east, by the Calder?
An actual local might come along with better advice
