What is the easiest way to find out how patients found our practice and keep track of referrals from other patients and doctors? Can we reward people who send us patients?

It's industry practice for physicians to inquire about the source of referral during the new patient registration process. I would encourage you to include a question on your new patient registration form asking how they found you. Offer patients a few options such as yellow pages, referral from a patient in the practice (leave room to provide the patient name), or referral from a physician (again, leave room to provide a name). 

You are not allowed to provide anything of value in exchange for a referral, but I would suggest that you send the referring patient or physician a thank you letter. Letter writing is a lost art, makes a huge impression, doesn't cost you a dime (well, 37 cents), and steers you clear of anything illegal.