The Imperial Palace is nice, but I wouldn’t really consider it necessary for one day in Kyoto. I believe you have to make a reservation, too. More traditionally, I would suggest:
Nijo Palace (gorgeous paintings, an excellent audio tour, and genuine “nightingale floors”
Kinkakuji (golden temple, very impressive looking but not much historical merit)
Ginkakuji (silver temple, gorgeous wood buildings not actually silver, but beautiful moss
garden, although crowds can ruin it a bit)
Kiyomizudera (THE temple in Kyoto, the one everyone knows, absolutely gorgeous especially if you visit at sunset, on a hill up a cool little kitschy tourist street, beautiful views across Kyoto and big enough grounds that you can find a little bit of peace)
Fushimi Inari temple (the one you see in, I think, Memoirs of a Geisha–hundreds upon hundreds of bright orange tori gates snaking up a mountain, the only Shinto shrine among the Buddhist ones I’ve listed so far)
Byodo-in (actually a short train trip out of Kyoto–about 1/2 hour, one of the most beautiful and famous temples in Japan, on the back of the 10 yen coin–one of the few remaining buildings from the Heian period)
Obviously there are a lot of temples here. I would recommend sticking to only a few from the above choices.
For example, you could go to Byodo-in in the morning, come back to Kyoto for lunch and try out the Philosopher’s Walk–you can see Ginkakuji and if you wander, you can discover a few neat little temples along the way–look out for a Shinto shrine with adorable statues of animals)
At night, I would advise going to Gion, if only for the atmosphere. You can walk beside the river, down the small street with extremely expensive restaurants and bars, and even check out the Gion Shrine (if you still have energy), which is quite beautiful at night. The road which runs to Gion Shrine has a lot of shopping, too.
You can also go to Fushimi Inari on the way back from Byodo-in, if you are really crazy for shrines…