A couple of things to do/check:
1. Make sure you have "Manually manage music and videos" checked under options for the device you want to transfer songs/playlists to.
2. You have to have direct access to the song files (either on you computer's hard drive, or a connected external storage drive)...
If you have iTunes match, the songs need to be downloaded to your computer, they will not transfer to your device from iCloud. So, if there is a little cloud icon next to the songs you want to transfer, you will need to download them first.
Now you should be able to drag and drop to your device. You can use the traditional sidebar, or simply grab the song/playlist and drag it to the right side of the iTunes window and a small side window will slide open with a list of available devices you can drop it in.