I reached an intermediate level of classic piano playing.
If I was to start playing again, how should I start, what things should I try to concentrate on? Or should I just learn again as if I was a beginner?
I would ask myself what kinds of music I would like to play now. Many piano players learned piano at a young age because their parents want them to play. Now you are at a point to want to do things for your own. If you like to be able to pick up songs and play by reading music, you should definitively start reading music again. If you like to improvise and play by ear, you can sit down on a piano and start "messing around" with a few keys and slowly add in keys and melody to start composing. There are many good courses out there to help you get to your goals.
Here is a few choices:
http://www.yokewong.net – Take your piano playing to the next level.
http://www.sightreadingtips.com – Improve on your sight reading skills.
http://www.playpianotips.com – Piano improvisation etc