It's possible to cycle in to the city centre from the south side pretty much separated from traffic all the way.
There's the South West Cycle route, fully separated down Shields Road (and quiet routes to it), along Scotland Street and down West Street to the Zig Zag brindge (don't know its real name
). There is also a route down Victoria Road.
Once across the Zig Zag bridge, there is a quiet road up to Waterloo Stteet, where there is again a separate cycle path. You can go through bus gates/bus only lanes on your bike and along Gordon Street (and West Nile Street? against the traffic) Alternatively, there is a cycle path along the north side of the Clyde/the Broomielaw.
There is an app called Bikehub which you can use to look for quietest/flattest/quickest routes.
As to where to "park" your bike, I'm not sure. There are plenty of proper cycle racks/supports (the nose in ones aren't as good as they can bend your wheels if the bike gets knocked over) but do get a good lock, preferably two, one a D-lock, if you're going to leave it there all day.
I cycle regularly from Pollokshields to the West End, over Bells Bridge and then the overpass from the SECC and there are only a couple of places where I come across traffic: crossing Argyll Street and Sauchiehall Street. Rest of the way is all quiet back roads.