How do you get a program to load earlier when you log in to Mac OS X?

The only way I can think of is to add the Logitech Control Center app in the list of app which launched at startup.

Preferences < Accounts < Login Items

But considering the fact it takes your Mac about a minute to load this app, I'm guessing the app is already in the list.

The fact that you have to wait a minute or so means your Mac is still booting (and loading all the software). I can't think of a way to increase this speed beside adding a faster drive or adding more RAM. But that's quite radical (and expensive) just to make your boot process a tiny bit faster.

An other thing you can do to speed up the boot process is to limit the apps your Mac has to open automatically (so it will get to the Logitech Control Center faster) and to clear your desktop (so you Mac don't have to waste time and RAM by loading all these icons and shortcuts).
This and more tips, are listed in this (rather old) article.

I assume that you check the "reopen all windows upon next login" window when you shut down. I have had the same problem.

The workaround is to disable reopen on login. Then the login items will be loaded much faster. I'm not sure if this is a bug and if Apple is working on it. But I'm sure that this is in fact your problem if you rely on a 3rd-party-app to use your mouse properly.

However, you should also check the settings in System Preferences → Mouse → Tracking Speed.