"I understand that we should only put stuff in there for the particular project that we are working on."
Yes this is to keep the library as small as possible.
"am I supposed to move this library to the "cloud" and how do I do that?"
You need an account with Dropbox (https://www.dropbox.com/) or other cloud storage provider. If your library is bigger than 5 GB (it probably is) you will have to pay for the extra storage you will need. You both need a fast internet connection else it will take many hours to upload.
"He also needs to access clips and events from my laptop for a different movie project he's working on."
I would start with just one project and an event with just the clips needed for it. This will be challenging enough. If you need to send him clips for use in his project the easiest way would be to email them to him using Apple Mail's Maildrop with which you can send files up to 5 GB. See: Mail Drop limits - Apple Support
"And are you saying that after one of us edits the specific project that we need to save it to the "new library" so that when the other person logs on to iMovie the revisions/edits will be there?"
If each of you have your the library on your local Dropbox folder it will be updated automatically when either of you makes changes - but it will take time and you need to make absolutely sure that the library does not change while either of you have iMovie running. In fact to avoid this risk (which could corrupt irretrievably the whole library) it would be safer for each of you to work from a copy of the Dropbox library that you manually update.
Although all of this is possible I have never actually tried to do it myself and from your questions I'm honestly inclined to advise that it is too ambitious - there are too many ways that things could go wrong.
Geoff.