Of course your choice of pubs to frequent will depend entirely on what you are looking for, but many visitors are looking for music, and I found several good places for that.
Some of my favorite pubs include a trio at the end of Stockwell Street: the Victoria Bar ("the Vicky"), Clutha Vaults, and the Scotia, all of which are quite old and established, and have great music nights. My favorite evenings at the Vicky are Tuesdays, Thursdays and Saturdays. They also have open sessions on Tuesdays and Fridays, where anyone can play (traditional Celtic stuff only on Fridays). The Clutha Vaults has blues on Fridays, country and blues on Saturdays and great jazz on Sundays. The Scotia also has live music Wednesday to Sunday. Otherwise, if it's not the music you are after, they are just great places to salve the thirst pangs, and you can divide your time between all three of them. To get to any of these from St Enoch Centre underground station, walk east along Argyle Street (all pedestrian friendly) and turn right on Stockwell Street.
Other good pubs for music include Molly Malones, on Hope Street in the centre. This is an Irish-themed pub that has live music 7 nights a week, and also serves food 12-8.
Another pub I enjoy is Uisge Beatha (pronounced, basically, "ooshga bayha"). This is the Gaelic word for whisky, and literally means "water of life". The male bar staff all wear kilts and the place is usually crowded. On Sundays they usually have open sessions, and Monday is quiz night. It is on Woodlands Road in the West End. A larger nearby pub suitable for heavy drinking is Hogshead, which is also on Woodlands Road, where it meets Gibson Street.
For more traditional music, try the Lismore and Ben Nevis. The Lismore has a session on Tuesdays and music again on Thursdays and Sundays. Ben Nevis has open sessions on Tuesdays, Wednesdays and Thursdays. The Lismore is on Dumbarton Road and Ben Nevis is on Argyle Street.
For something totally different from these more traditional Scottish pubs, try Revolution, an vodka bar on Renfield Street. This place is a bit clubby, but serves excellent (if expensive) vodka drinks. It has a hip feel to it without being snobby.