The only real consequence is the drive must be connected to listen to the music. But as long as you know that it won't be a problem for you.
It's easy to do this from right within iTunes.
1. Connect the external HDD.
2. Open iTunes Prefs > Advanced. Where it says "iTunes media folder location" click Change.
3. Specify your external HDD and click OK.
4. Now, since you've said you want to move all the music files to the external HDD click Music and highlight all of your tracks.
5. Right-click (or Control+Click) in the list and choose "Consolidate Files." This will copy all the music files to the new iTunes library location on the external HDD.
Once you are satisfied the process was successful you may delete the folders containing the music from the Music folder. However, do not delete the .itb, .itl, or .xml files you find there. Those need to stay where they are. After this, with the external HDD connected all new music will be download to that location. However, always remember, that when the external HDD is not connected the iTunes library location will revert to its default location which is in the Music folder, and if you purchase any music that is where it will download to. Other than those things you'll be fine. I've run my iTunes library off an external HDD for years.
In fact, for one reason or another, I got interrupted, so apologies for the delayed response to this helpful advice.
My only question arises at stage 4. If I click "Music" (I assume in the main iTunes window, as there is no other option), nothing happens. In other words, I get as far as designating my hard drive as an iTunes media location but, because the Preferences window is open, I cannot get into the main iTunes window to select Music (and then highlight all tracks).
The other question I have is about the effect of "Consolidate Files". I haven't got to this point, because of the hold-up at the earlier step. However, I would like my music to be ordered just as I left it: i.e. in files and subfiles (e.g. suppose I had a recording of, say, Abraham Lincoln's piano concerto: I'd like it to be in the original three movements, as originally stored, with all the other stuff filed the same way.
Sorry to sound thick. I'm sure I'm missing something obvious and strarightforward, but I don't quite get it!