I spent my first day in Tbilisi exploring the left bank of the Mtkvari river, which according to the guidebooks (Lonely Planet and Bradt) had nothing to see. Which I would disagree with, as I thoroughly enjoyed my day exploring, in a slightly lost fashion, the area. Even more perhaps than my subsequent day in the Old Town seeing the better known sights.
I often find getting slightly lost one of the best ways of exploring a new city. I had a map and a general idea of where I wanted to go, but was quite happy just to wander the streets and see what found. Which in this case was a lot of rather evocatively derelict, or at least somewhat knackered, buildings.