As Nai said, this is simple and iTunes is designed to make it simple, but... as with all things simple, there are simple ways of messing it up so I would take the time to do things step-by-step.
Few caveats to be aware of.
I'd firstly check in your advanced preferences to see if it is checked off that iTunes will automatically:
1. Manage and organise your content
2. copy files to your library.
If you aren't too fussy about how things must be organised, iTunes will sort it all for you into artist, album etc. With (2) checked, it means all content is stored under the iTunes content folder.
If (2) isn't checked, while the content shows up in iTunes, it might not be under your iTunes content folder.
Having iTunes do all the organising will make the move nice and easy.
Next caveat:
I'm doing this the easy and simple way, but first have a look at your iTunes folder in My Music you have two things: iTunes Database and iTunes Content.
It's the Content you want to move, not the Database. If you copy and paste the whole folder then you will incur the Wrath of the Apple God of "Why didn't you just use our Consolidate feature now I have to have an exclamation mark against every track you numpty". Trust me, he's a very busy god.
Just to be sure, if you go into your "My Music" folder, there should be an "iTunes" sub folder. In there is:
“iTunes Library.itl” or iTunes Library Database and iTunes Music Folder (the content)
The database is just that and isn't a large file at all, do it the "Apple" way and all should be sweet.
Moving:
1. Go to advanced preferences and change the "iTunes Music Folder" path to your new location.
2. Then go to File, Library, Consolidate Library.
3. You will get a warning that this cannot be undone, your last chance to back out if you're nervous. But be tough, press ok/continue or whatever.
4. Sit back and wait/panic.
5. Tis done.
Final caveats:
This process has only copied and not moved your content. While it has updated your database (remember different thing) as to where the songs are now kept, the original content still exists on the current drive. You can delete this old content folder, but I would do a few checks first.
Simplest is pick a few tracks at random (do a shuffle or genius playlist) and for each one go through the file, get info option (or right click). There's a summary tab and it should tell you the new location. If the new location is there for all, then wayhay!
If you're feeling confident in iTunes' simple process you can now delete the content folder, but make sure it is just the content folder (see above), leave the database where it is as there's no need to shift this. Some nervous souls like to also do a back-up back-up of the contents to be sure to be sure. Me: I live on the edge, so once I've confirmed a few tracks are ok, I delete.
Final Final Caveat:
Always, always, always make sure that the external drive has been recognised by your computer before you open iTunes. Get trigger happy and open iTunes first and you're in for a world of pain and the other Apple God of "I told you not to do that and now I'm going to lose half the tracks and duplicate others while messing everything else up too."
Apple Gods are vengeful Gods.