Kyotoites think of their city as a Japanese Paris of sorts and it shows in a French touch in much of the bar scene. Italian style is also popular.
For fashionista Izakayas Pontocho is still the right place, Imadegawa street has a few beautiful bars/cafes. Or you could get up to Kurama, buy some rice wine from the village shop and sip it while bathing in the outdoor tubs.