Here's a test you can do, to trouble-shoot the problem with the delay when opening iTunes, to narrow down and isolate possible causes.
Create a new OS X user account in System Preferences Users & Groups pane. The "Test User" account can be a Standard (or Administrator). Log out and log in to the new user account. Do not use Fast User Switching.
Set up the new user account like your primary account, linking it to your Apple ID. Run iTunes, and sign in using your Apple ID. This is a new iTunes library at this point. If you have songs purchased from the iTunes Store, you can use the Purchased screen to add some of your past purchases to your iTunes library
Download your past purchases - Apple Support
When you quit and relaunch, do you experience any delays? If you do not, the cause of the problem is something in your usual account, such as your iTunes library database or third-party background processes that run when you log in. If the same delay does occur in the new user account, then the cause is at the system level (or maybe something related to your Apple ID in this case); that would be the OS X system, or the overall iTunes application.
One thing to check for with iTunes (at the system level) is "plug-ins." Look here (using Finder)
[startup disk]/Library/iTunes/iTunes Plug-ins/
for any iTunes "extensions" from third-party developers (not Apple). I have only one item there; it is called Quartz Composer Visualizer.bundle. Doing a Get Info on it shows it's from Apple. If you have anything there that is not from Apple, and it's something you don't need, remove it. See if the delay goes away.
Another thing you can try (at the system level) is to reinstall iTunes. Go to the iTunes download page
https://www.apple.com/itunes/
NOTE: If you are intentionally using an older version of iTunes, this gives you the latest version.
Download the installer and run it. You should always have a backup of your iTunes library data (along with the rest of your user data), but reinstalling the iTunes application does not affect your iTunes library data. (Start using the built-in Time Machine feature, if you are not already, for automated backups of your entire system.)
If the new user account test points to problem at the user account level (not system level), one place to check is your Login Items. This is at System Preferences Users & Groups pane Login Items tab. These are processes that start up automatically when you log in to your account. One of the items is probably iTunes Helper, and that's fine. If you see any other item(s), and you don't know what it is or what it does (or it's for something old that you don't use anymore), you may want to remove it (them) from the list (select it and click minus sign). Don't remove something that you are currently using. If you made a change, log out and log in (or restart) and see if there's any difference with iTunes.