How Do I Add a Song to a Playlist?

Several alternatives:

• When you're viewing My Music, if you drag a song (or selection of songs) to the left edge of the screen, the sidebar will open and show your playlists - drop the song(s) on the playlist you want to add to.
• If you click on Playlists (top center of the iTunes window) the sidebar will be displayed, showing all your playlists. If necessary, click on the Music entry at the top of the sidebar to show your complete library. Drag and drop operations then work as they did in prior versions.
• Right-click on any song or selections of songs, select Add to Playlist and pick the playlist you want to add to from the resulting submenu.

Option 1 applies when you're viewing your complete library via the My Music button at the top of the iTunes Window:

The sidebar behavior has actually changed in iTunes 12.2 - it will become visible as soon as you select and start to drag in any direction (12.0 and 12.1 had the "drag to left edge"):

You can then drop the selection onto your choice of playlist:

To have the sidebar visible at all times, just click on Playlists at the top of the screen:

Works for me whether My Music is in Albums or Songs view:

I've checked all three systems I have access to right now (Win7/32, Win8.1/84, Win8.1/32 - all running 12.2.0.145) and this behavior seems to be consistent. Yet you're just seeing the little bar with the disc/track nos. & song title plus the red/white circle, without any appearance of a sidebar to the left of the screen? With 12.2 being new and not exactly bug-free, this may be something that, if others can confirm it as reproducible, should be reported to Apple's iTunes feedback page.