Here's what I consider the most worthwhile sights for the first-time visitor to Kyoto. I'm judging by what my students enjoyed most when I used to take them there.

Kiyomizu-dera. As a temple it's not that different from the others, but the walk up the hill takes you through a well-preserved neighborhood, and the view of the city from the "terrace" is great.
Kinkakuji (Golden Pavilion) and Ryoanji (the raked sand place): These are so famous that you don't want to miss them, but unless you are deeply philosophical, you won't want to spend that much time there
Ginkakuji (Silver Pavilion) and the Philosopher's Walk: Start at Ginkakuji and walk south along the stream. It's a pleasant, wooded walk and you may or may not want to stop in at the other temples along the way. Eventually the walk leads you to the area of the Chion-in, but unless you have a lot of time, I'd skip that and go straight to Gion and poke around in the stores.
If there's traditional dance show at the Kaburenjo theater, you might want to take it in.
Nishikikoji Market: Across the river and a few blocks inland from Gion is Kyoto's central market arcade. Challenge yourself to see how many of the foods you can recognize.
Nijo Castle: a genuine feudal castle, well-preserved, gives you an idea how the samurai really lived
Nishijin Textile Museum: I was surprised at how much my students enjoyed this. You can see demonstrations of how the famous Kyoto brocade is made, and there's a kimono style show every couple of hours.
Pontocho after dark: This is the prime geisha district, full of traditional buildings, most of which function as private clubs. You are likely to see geisha and maiko making their rounds from club to club to entertain at private parties.

In summer I would add the beer gardens along the Kamo River, since Kyoto has some of the muggiest, most oppressive summer heat anywhere.

I wouldn't bother with the Imperial Palace tour, even though you're supposed to be privileged as a foreigner to join a tour without an appointment. The problem is that you don't actually see anything. The guide just walks you through the palace grounds and points out non-descript buildings that you never get to see the inside of, even though no one lives there any longer. There's a book for sale at the gate that shows you what the interiors look like, and if you're interested, buy that. It shows you much more than the tour does.

The best way to plan a tour of Kyoto is to get out a map, preferably one with the bus routes superimposed. The bus ticket center in front of Kyoto station sells 24-hour transit passes that are good for the buses and subway.