Hi Lila, welcome to CC!
I live in South Lanarkshire myself, Rutherglen.
It's dire for inexperienced cyclists, segregated facilities are few and far between.
I would start with taking the kids to Chatelherault country park: there is now a cycling path that takes you to Strathclyde park from there.
I guess the kids can only do weekends and school hols.
If you can get to Morrissons Cambuslang, just across it there is the entrance to the Clyde river path, takes you all the way into town.
CamGlen Bike Town ( find them on FB or Google their website) are a cycling hub in Rutherglen, they organize beginners rides on weekdays, free parking available.
@flyingfifi is one of their ride leaders.
Freewheel North in Glasgow Green also do beginners rides (kids welcome) on a Tuesday from 10.30am to around 1.30: you can hire a bike for a pound donation, this is a short walk from Bridgeton station, free parking available.
There are a few ladies only groups in Glasgow, the Belles on Bikes, the Glasgow Gals, Breeze, all have web and FB presence.
Anymore info, just ask!