I assume they are not both on the same network. If this is the case, I find the USB flash drive to be the easiest. I use it all the time to transfer files from work to home. I don't know what I did for all those years without it. Watch the weekend sales flyers, there is usually someone offering a deal on them (bestbuy, circuit city, office max, staples, etc.)
In a pinch, I have emailed files to myself (send the email from the old computer and then open the email on the new computer)
If your talking about files such as word documents, the OS level doesn't matter, it is the application level that counts. Most apps continue to support older levels.