I am going to Japan
 

I am going to Japan with a female friend form Dec 21 to Jan 2. We want to go to Tokyo, Osaka, and Kyoto. So I think the plan is once the plane arrives in Tokyo we will take the train to Kyoto or Osaka, and stay at a hostel for 3 days in Kyoto and 3 in Osaka then stay in Tokyo till we depart on Jan 2.

I could use some advice about where to stay and where I can find more information on hostels and things to do there.

So after a 12 hour flight you're going to take the train to Tokyo or Shinjuku station (about 1 or 2 hours), then after finding the shinkansen platform and getting tickets (30 mins if you're lucky), waiting for the next train (30 mins to an hour), you will get on a train for another 3 hours of traveling?

That is like 20 hours of traveling. Why are you not flying directly to Kyoto and leaving from Tokyo? I forgot to add the time you will spend going through immigration and getting luggage off the plane (2-3 hours). If you can get a flight directly to Kyoto or one that flies through Tokyo to Kyoto, you would be a lot better off.

Depending on what time you land, immigration in Osaska (Kansai Intl Airport) is only 15-45 minutes.

I agree, spend at least a night (or 3) in Tokyo before heading to Kyoto and Osaka. But be prepared for the country to mostly shut down from 12.31 to 1.3.

And there is lots of great stuff to see in Tokyo, Osaka and Kyoto, but without having the foggiest idea what you are into, I am not even going to venture a guess. All of these cities literally have thousands of websites dedicated to describing what they offer for tourists (though you are going at a kind of crummy time). Ask about specific places, and I bet you'll get some educated responses.

lets just say X= hometown

take this destination, it will say you time

X = KIX ( Kansai International Airport )
Narita (Tokyo) = X

when you land at Kansai, take the train for Kyoto, that will take you around 1.5hours - 2 hours travel

actually you don't need to transfer hotel from osaka and kyoto , this 2 cities are just 30-40 minutes away from each other by express train. I recommend staying in Osaka cause its in the middle of everything, like if you feel to go to Kobe, nara or wakayama

if you dont have the jr pass, use hankyu line going to kyoto ( kawaramachi ) cause its the cheapest ( 390 yen ) compared to 600++yen of JR. And the Kawaramachi station of hankyu is already in GION, so you will be in the middle of everything Kyoto unlike in JR Kyoto, its in the middle of nowhere :D
when in kyoto used the one-day ticket bus ( 500yen ) you will be able to use all the city for the whole day this will get you anywhere in kyoto. you can get the map at the station ( JR or Hankyu )

if you are planning to tour osaka for a day, i recommend getting the 800yen one-day ticket for all the subway system, this will get you anywhere in osaka. subways are expensive so this is the cheapest way getting around the city. there are also 2000yen or 3000yen ticket which would allow you to visit some establishment and get discount on selected restaurant.. but nahh i don't recommend it... :D

enjoy your trip :D