How do I establish a strong rapport and build trust with my patients as a family nurse practitioner?

Take the time to actually talk with your patients. Actually listen to what they are telling you. Allow for questions and answers. In other words, get to know them as if it were your friend or family member.
I am an ER Nurse but also an ESRD/Renal Transplant Patient. So I've been on both sides of the bed rails. There is nothing worse than having your provider, whom you've been seeing for quite sometime, come in and ask you a million questions about your chronic condition(s) that they should already be aware of. Read through their charts. I'm not suggesting you write a double spaced 12 page book report on your patients. It is just important to listen to and get to know your patients.