How do I create a button on a form that will save the current record, copy some of the fields to a new record, and redisplay the new record in edit mode?

Glad you are getting close.  We will have you being an API whiz before you know it.

You cannot populate a lookup field.  So what you need to find is the field [Related Parent] for where the lookup came from - ie the "reference field' on the right side of the relationship and populate that field.  Then ALL the lookups for that relationship will come down automatically.