One-day-trip Tokyo-Kyoto-Tokyo

We will be visiting Tokyo for a week in september. Now two of us have been to Tokyo once before and love the city. We are planning to take one day trip to Kyoto for visiting the ancient royal city.
Questions:
- Is this easy done with the chinkansen and what is the timetables? How much will this cost with the faster trains?
- What to see and what to do first when you only have one day in Kyoto? Most important sights?
- Shall we book this by an agency (and then who?) or can we manage this ourselfes?

Is this easy done with the chinkansen and what is the timetables? How much will this cost with the faster trains?

Yes, it is easy. Here you go with the details:
http://www.japan-guide.com/e/e2363.html

What to see and what to do first when you only have one day in Kyoto? Most important sights?

Here you go:
http://www.japan-guide.com/e/e2158.html

Note that a 7-day Japan Rail Pass costs about the same as a Tokyo-Kyoto round trip by regular shinkansen tickets.

Shall we book this by an agency (and then who?) or can we manage this ourselfes?

It can be done easily by yoruself. But pricewise, you might want to look into the shinkansen package deals by JTB, although they include overnight stays:
http://www.jtbgmt.com/sunrisetour/ex/

The trip from Tokyo to Kyoto is about 4 hours by shinkansen. If you're thinking about going there and coming back on the same day, that doesn't leave a whole lot of time in Kyoto, plus it would total to more than $200 per person for a roundway ticket. I recommend finding a cheap hostel or ryokan in Kyoto. You can easily book online. Stay there for one extra day and then return to Tokyo.
Hostel prices run from 1400 yen to 3000 yen while ryokans are 3000 yen to about 6000 yen a night.

The trip from Tokyo to Kyoto is about 4 hours by shinkansen.

Actually the trip from Tokyo to Kyoto by Shinkansen is between 2hrs 15 mins (Nozomi) and 2hrs 45 mins (Hikari), so you could easily do it in one day, leaving Tokyo at around 8am getting to Kyoto at approx 1130, and leaving at approx 1830 and getting back to Tokyo around 2100.

Actually the trip from Tokyo to Kyoto by Shinkansen is between 2hrs 15 mins (Nozomi) and 2hrs 45 mins (Hikari)

The difference in cost between Hikari and Nozomi for this journey is about 300 yen!

Take the Nozomi and have more time on your vacation.

The difference in cost between Hikari and Nozomi for this journey is about 300 yen!

The difference in cost can be considerably higherin case of Japan Rail Pass holders.

When i took the trip, i remember it being 4 hours.. maybe 3.

The difference in cost can be considerably higherin case of Japan Rail Pass holders.

That is true, but the original poster never said he had a rail pass. I don't think it can be assumed that everyone will get a JR rail pass. IMHO the value of the rail pass is debatable: it's worth it if you make lots of long trips but then you will never get the time to enjoy being in the places you went to see. If you factor time into the equation of cost, the original poster only has 1 week and taking a whole day just for travelling may not be worth it.

With a 7 day JR Rail Pass bought prior travel outside Japan:
First hikari from Tokyo lv 06.36 h arr Kyoto 09.20 h, next one Tokyo lv 07.06 h, further one always hour.36 h and hour.06 h.
Last hikari to Tokyo lv Kyoto 20.39 h arr Tokyo 23.19 h.
Without JR Rail Pass take nozomi shinkansen around 25 min faster than hikari and 8 to 10 trains within every hour.
In Kyoto buy one day Kyoto City Bus ticket at Yen 500 and visit Nijo jo ( shogun castle ) Ryoan ji ( garden ) Kinkaku ji ( Golden temple ) Daisen in ( garden ) Ginkaku ji ( landscape garden ) walk down Path of Philosopher and last Kiyomizu dera ( temple area ). Although I did it with a day visitor one day you can skip what you do not like for a slower visit.

I must have mistaken the time and money it takes to get to Hiroshima from Tokyo. Sorry about that. From there we went to Kyoto and then returned to Tokyo. It was an expensive trip. Sorry for the slight confusion.

That is true, but the original poster never said he had a rail pass.

But I recommended to consider a rail pass. It is likely to pay off in this case, especially if a round trip to Kyoto is combined with a few other JR trips.

I have been living in Kyoto and certainly I agree with Miko, you should stay at least one night to be able to enjoy just a little of Kyoto.
Anyway if you just have one day I would recommend you should visit Nijo-jo (Nijo Castle), where you can have a good glimpse of how was the life of the Shogun and then you could just head north and visit Kinkaku-ji. (golden pavillion), it is the most photographed place in Kyoto by far.

If you want to enjoy a little bit of the old atmosphere of Kyoto I would also recommend you other itinerary.

From the station take East and visit Kiyomizu-dera. It is a rare temple in Japan, since it is suspended in the air thanks to big wood pillars.

Kiyomizu-dera is good but it is better the walk from Kiyomizu- dera to Gion temple. It is an amusing walk though one of the eldest areas in Kyoto.

Once you arrive to Gion Temple, make sure you are there around five p.m. At that time you can just walk around Gion Corner district (just in front of Gion Temple) and you can have the chance of getting a Maiko (Geisha`s apprentice) walking or shopping to get the goods for that evening guests.... It deserve a walk¡¡ this is not possible to find in Tokyo- Believe-me.

Be sure not to confuse real Maiko with fake Maiko. "Fake Maiko" are girls who paid a studio about 8000 yen to get dressed up as if they were Maiko, and then with their own camera, a photographer takes their picture in front of a famous temple. You can tell that they aren't real by looking at their hands. If their nails are "Koda Kumi" style (a famous pop star in Japan) she certainly isn't a real Maiko. Also you'll notice a person in casual clothing guiding them around the area.

But you can also participate by finding a studio in Kyoto and pay the money to get your picture taken.