Nijo Castle, Golden Pavilion, Kyoto Imperial Palace.
No so obvious things:
Kiyomuzu Temple
You get to this hill-top temple by a long steep street lined with shops, which is interesting in itself (but could be tiring for the infirm). Lovely views from the temple at the top, especially at sunset. At dusk, they light lanterns round the temple. Entrance ¥ 300.
Gion
Wander around this area of Kyoto in the evening. Good restaurants, nice Kyoto atmosphere. "Gion Corner" has performances of traditional ceremonies and dance, at 19:40 and 20:40 every evening Mar-Nov (¥ 3,000).
Philosopher's Walk
A walk along the canal, past cherry trees, bridges and temples, made famous by the philosopher Nishida Kitaro, who liked to walk along this route and meditate. Many cafés and restaurants en route (although the ones just off the main beaten track are cheaper). Do it in either direction: from the Ginkakuji (Silver Temple) to Nanzenji Temple off Sanjo-dori, or vice versa. Particularly pretty (and busy!) at cherry blossom time.
Pontocho-dori
A narrow picturesque lane, crammed with restaurants (rather expensive, but nice to look at!) good for an evening stroll. A narrow canal (Takasegawa) runs parallel to it all the way from Sanjo to Shijo-dori. All the little bridges, bars and restaurants look great at night.
A trip outside Kyoto: boat trip Kameoka - Arashiyama
Get the train from Kyoto to Kameoka, follow the signs for the river-boat. You sign up and pay, and wait for your name to be called to actually board a boat. The trip along the Hozu River is particularly good when the trees are in their autumn colours (Red maple) but avoid the weekend crowds. When you get to Arashiyama there's time to explore.
Or simply take the train all the way: from Kyoto Station to Arashiyama, take the JR Sagano Line (San-in Line) to Saga-Arashiyama Station (about 15 minutes) and it's about 5 minutes walk into the centre of Arashiyama.
Wear sensible/walking shoes. For the footsore, people distribute leaflets in the streets of Arashiyama advertising "foot-onsen" (some adverts might be for "love hotels"... not difficult to spot the difference!)