You could park in Strathclyde Park, junction 6 end in one of the free carparks. Make your way through the park west towards Glasgow. Come out the park at junction 5 of the M74, massive roundabout. Stay on the pavement / cycle path. Follow it round left crossing the onslip south bound to the next junction, the offslip from the M74 onto the roundabout northbound. Cross this road at the lights and keep an eye out for a path heading off to the left. Take this path that disappears into trees and bushes but follows the road, this will take you under the bridge along the River Clyde and over a raised walkway. At the other end you will come out at Bothwell memorial.
You’ve now got a choice of heading across the road and left, cross the bridge and head right along the cycle walkway to Blantyre. From Blantyre you can follow the A724 straight into Glasgow or divert left at Drumsagard onto country roads further up in the hills. (You could if feeling fit get to Cathkin Braes this way). Nearer Glasgow you come in through Bridgeton, Glasgow green and the River Clyde. This is my commute. Fairly safe even at night in winter.
Or you could head right up the hill towards Bothwell, pass through to Uddingston and take London road into Glasgow. London road is a 2 lane, 30/40 mph not too busy carriageway through Parkhead and the Sir Chris Hoy Velodrome. Take you straight into Glasgow at the Barralands.
If you decide to stay for a few beers one night just get the train to Hamilton and freewheel down the hill to the park, cutting through the Palace grounds.
