Really depends what kind of pub you're looking for. If you're a fan of whisky then The Grill on Union street, if you're looking for a younger crowd, then Belmont Street (just off Union Street) has a few. Civilised drink with your missus, then Under the Hammer is supposed to be good since it's been done up.
If you're proper old school then head to Portals or The Carlton in the Castlegate and drink with a load of old men.
If you want a fight with your drink the Justice Mill Lane was always the place to be.
And there's a new wine bar called Faffless across from M&S. (I've never been)
Drummonds for music and a late drink. Some 'interesting characters' in there as well.
And of course, there's the Brewdog pubs. Castlegate or Gallowgate.
ETA: I should add that I rarely go out in town these days so the younger folk on here would be better placed to give you advice.