You most likely will not be able to "pull those over" from your other phone by somehow connecting them via cables or downloading something to a disk. The ringtones that are installed in phones by default are in the phone's ROM most of the time, which is Read-Only-Memory, and exist regardless of what you do.
If the ringtones you speak of are just from songs, there are free apps available on the Marketplace for you to download that take an entire MP3 song that you have on the Droid (slap it on your MicroSD card), and then allow you to splice it to a ringtone-sized clip to create custom ringtones. In this case, if your ringtones from your LG are songs, find those MP3s that you own and rip them.
If you bought the ringtones, once again you may have trouble bringing them with you. A lot of people get tripped up on this - they spend a bunch of money on legit ringtones, but then come to find out that you can't take them with you to a new phone. Usually when you buy that ringtone, its for that phone specifically.