Must-go/see shrines in Kyoto

I am travelling to Kyoto and would like to visit some temples and shrines. However, there are so many I can't possibly visit them all, so which are the must-go ones?

I recommend the ones with three red dots, followed by the ones with two red dots:
http://www.japan-guide.com/e/e2158.html

I agree with Uji's suggestion, but if you can state the exact dates you will be in Kyoto sometimes the recommendations could change due to temple markets, cherry blossoms, and autumn colours.

Some of the Kyoto suggested itineraries are useful too:-
http://www.japan-guide.com/e/e3950.html

Enjoy your trip!

I'm going to Kyoto from 12-15 Dec 2012. So it'll be winter.

When will you be traveling? Sometimes a temple/shrine may fall off the radar during one time of the year but becomes a "must visit" during other times (ex: when flowers are blooming).

In general, I think the World Heritage Sites are also worthwhile (perhaps with the exception of Nishi Honganji Temple).

In terms of temples, some of my favorites are (the 1st 4 are probably better for a 1st time visitor, although Shisendo and Sanzen-in do not disappoint either. They're just less typical. The others I like but don't necessarily recommend unless you have certain interests):
* Daigoji Temple
* Sanjuusangendo
* Toji Temple
* Kinkakuji Temple
-Shisendo Temple
-Sanzen-in Temple
-Jingoji Temple
-little Gioji Temple
-Yoshimine-dera
-Rokuharamitsuji Temple
-Yogen-in Temple

Some of my favorite Kyoto shrines:
-Shimogamo Shrine
-Kamigamo Shrine
-Fushimi Inari Shrine
-Heian Shrine
-Jonangu Shrine
-Umenomiya Shrine