Hi
@dianelc and welcome.
I live in Rutherglen, which is the town before Cambuslang.
Be aware that the path along the Clyde from Rutherglen to Cambuslang is closed until further notice because they are building a new bridge.
There are no diversions in place, as far as I know you would have to go on the main road, which is a fast and busy road.
After Cambuslang I believe you have to get to Hamilton (on another fast, busy road with painted on cycle lanes), then go through Chatelherault park to join the cycle route.
All in all the route from Glasgow to Hamilton is a bit hairy and all uphill. Even to enter the park you need to turn right after a steep descent, with incoming traffic on a blind bend.
If you're riding with a 10 year old, I would advise the canal route from Glasgow: it's a bit dull, exposed to the elements, rough in places (not suitable for road bikes) but safe - well, unless you fall into it!
I would strongly NOT recommend the route
@boydj suggests if you are inexperienced.
From Paisley to Glasgow the NCN7 meanders a bit, but is safe and well sign posted, only you need to watch out for unexpected twisty turns.
When are you planning to do this?
I could scout the Hamilton - Bathgate for you, but Glasgow to Hamilton will always be difficult with a 10 year old or if you're not used to riding in traffic.