Mode of transport

We are travelling with a 8-y.o child. What would be the cheapest and most convenient mode of transport from Osaka to Kyoto and from Osaka to Tokyo? Can i make the arrangement when i'm there in Osaka?

Most convenient mode is not necessarily the cheapest one. There're several train lines connecting Kyoto and Osaka and it depends on which part of the cities you are staying at. But from Osaka to Tokyo, I recommend using shinkansen (bullet trains).

which part of Kyoto / osaka i should stay to maximize my 4-day trip. what are the sights / places i must visit in Kyoto & Osaka? How much is the shinkasen fare from osaka to tokyo?

for all train fares, schedules etc. check www.hyperdia.com
the most logical places to stay would be near JR Kyoto station and near JR Osaka station. or you can stay all the time in Kyoto (cheaper accomodations near the station) and go to Osaka by train, it isn't such a big deal. You really should get a 7 days JR pass then. Check about passes on the general japan-guide site under transportation.

--------We are travelling with a 8-y.o child. What would be the cheapest and most convenient mode of transport from Osaka to Kyoto and from Osaka to Tokyo? ----------

Japan Guide here has a info for Highway buses and it's fares. Night Bus would be good way to travel if you think your 8 year old can sleep thru it.
Kyoto to Tokyo is $42.50 on a express bus, takes 7 hours.
$135.00 on express train, takes 2.5 hours.
http://www.japan-guide.com/e/e2366.html

sorry about that.
Correct info is...
Osaka to Tokyo is 3,500 yen on express bus and takes 8 hours.
On a express train is 14,000 yen and takes 2.5 hours.

How to travel between Osaka and Kyoto:
http://www.japan-guide.com/e/e2363.html?aFROM=2157_2158

How to travel between Kyoto and Tokyo:
http://www.japan-guide.com/e/e2363.html