For a fun early evening drink, visit Maruyama park.
Go to nearby Gion one evening and try a Kaiseki. These are small, intimate restaurants where you sit at a counter and watch the chef at work You do not choose the menu but simply get presented with 10 courses (or so) of the chef's choice. These range from sashimi and bamboo through to live abalone, cooked in front of you!
We visited two Kaiseki places. Sagakawa and Gion Karyo. The former was more intimate and more of an experience; the latter was better in terms of food.
For Soba, Ramen or Udon, look no further than Honke Orawiya.
For a fantastic wagyu beef lunch, go to the number one restaurant on trip advisor - Otsuka.
One evening, visit the atmospheric Pontocho St (near Gion) and eat at a Sukiyaki restaurant. We loved Sukiyaki Iroha (there are two addresses - one at the end of Pontocho, which was closed, and one further up).