How do I get to use Font Book fonts in TextEdit?

If the Symbol font is active in Font Book (and it should be), then Character Palette should be able to do the job. Make sure that you open up the Font Variation arrow near the bottom of the Character Palette window, and pick "Containing selected character" from the little Collection menu down there.

The Symbol Regular rendition of the character will appear in the list, click on it and press the Insert with Font button at the bottom right.

Or, you can type the alpha with any old font, select it, press apple-T in TextEdit to bring up the font inspector, and set it to use the Symbol font.

 

Font Book is really there so that individual fonts can be installed, checked, activated and deactivated systemwide. It's not made for dealing with individual applications, the apps do that.

You don't actually need to use Character Palette for this. If you want to use lots of Greek letters, you can turn on the Greek Polytonic input method in the International section of system Preferences. You can peek at the Keyboard Viewer to see what this does.

To make switching input methods easier, you can set up a hot key to switch them in the Keyboard shortcuts pane of the Keyboard & Mouse system prefs (look for the Input Menu section). Unfortunately, the brilliant human interface people at Apple decided to assign the exact same keystrokes to this as they later assigned to Spotlight, so you will need to think up new keystrokes for one or the other.