List of free online courses:

Family History/Genealogy - Introductory
FHGEN 68 — Finding Your Ancestors
FHGEN 69 — Providing Temple Ordinances for Your Ancestors
FHGEN 70 — Introduction to Family History Research
FHGEN 80 — Helping Children Love Your Family History

Family History/Genealogy - Record Type
FHREC 71 — Family Records
FHREC 73 — Vital Records
FHREC 76 — Military Records

http://ce.byu.edu/is/site/courses/freecourses.cfm

BYU also offers a Certificate Program in Family History...

The Certificate Program in Family History (Genealogy) is designed for those persons not desiring to complete all of the requirements for a degree or who can't return to campus to do so. The certificate program provides a solid background in fundamental family history research principles coupled with specialized genealogical training in a particular geographical area.
Requirements

The certificate requires 18 hours of course work: HIST 400, The Family and the Law in American History (3 hours); HIST 421, English Language Handwriting and Documents (3 hours); HIST 482, Professional Paths and Credentials in Family History; two source courses (6 hours), see list below; one elective course (3 hours), see list below.

College credit is granted for each course successfully completed to fill the certificate program requirements. The certificate is not to be viewed as a college degree. For students not interested in credit or the certificate, all courses may be audited. The courses are the same in content, but no credit is awarded and no final examination is given.
Required Courses

HIST 400 The Family and the Law in American History (3 hours)
HIST 421 Section 1, or HIST 421 Section 2 English Language Handwriting and Documents (3 hours) HIST 482 Professional Paths and Credentials in Family History

Choose two courses from the source courses (6 hours), and one course from the elective courses (3 hours) in either the American or British option.

North American Option

Source Courses

HIST 403 U.S.--Midwest Family History Research

HIST 404 U.S.--Southern States Family History Research

HIST 411 Latin American Family History Research

Elective Courses

HIST 378 American Family History

HIST 432 Oral History Interviewing and Processing (currently not available)

HIST 433 Writing Family Histories

HIST 481R Section 1 (3 credits), or HIST 481R Section 2 (1 credits) Directed Research in Family History

British Option

Source Courses

HIST 409R (1) Germanic/Slavic Family HIstory

HIST 413 England/Wales Since 1700 Family History Research

HIST 414 Scottish Family HIstory Research

HIST 415 Irish Family History Research

Elective Courses

GERM 490R(3) German for Family Historians and Genealogists

HIST 319 The Family in Europe

HIST 322 England to 1689

HIST 323 England Since 1689

HIST 390R (1) Special Topics in History

HIST 432 Oral History Interviewing and Processing (currently not available)

HIST 433 Writing Family Histories

HIST 481R Section 1 (3 credits), or HIST 481R Section 2 (1 credits) Directed Research in Family History

* The seminar course, HIST 481R, may be completed in one of the following two ways.
1. If you are attending the Family History Conference held at BYU, enroll in HIST 481R, Section 2 (1 credit). You will also need to register for the BYU Genealogy and Family History Conference for two credit hours. The conference is held on the BYU campus, usually the first week of August. The Department of Independent Study does not sponsor it. All correspondence should be sent to Genealogy and Family History Conference, BYU Conferences and Workshops, 136 Harman Continuing Education Building, Provo, UT 84602.
2. If you are not attending the BYU Family History Conference enroll in HIST 481R, Section 1 (3 credits). In order to earn credit for this course, you must accrue 32 hours of Genealogical activity. At least 20 of the hours must be approved seminar sessions. Any additional Hours needed to total 32 hours are obtained by doing Genealogical research at a library with substantial genealogical holdings. Three seminars are currently pre approved for the course requirements: the BYU Genealogy and Family History Conference (see #1 above), the National Genealogical Society Seminar (for 5-7 17th N., Arlington, VA 22207), and the Federation of Genealogical Societies Seminar (P.O. Box 3385, Salt Lake City, UT 84110-3385). If you are unable to attend any of these seminars, you may obtain approval for seminar(s) in your area. You are responsible for locating these seminars in your area. Send Independent Study a detailed copy of the seminar program. We will inform you if it has been approved and, if so, how many hours will be obtained. If the number of hours is fewer then 20, you will need to attend additional seminars to satisfy the requirements (make the appropriate arrangements with your instructor).
To Apply

Send a letter of application to Roy Schmidt, Family History Certificate Program Coordinator, Independent Study, 239 Harman Continuing Education Building, P.O. Box 21514, Provo, UT 84602-1514. Please include the following:
Name
Address
Date of Birth
Social Security Number
Telephone Number
A paragraph describing your experience, if any, in family history/genealogy
A paragraph explaining why you want to earn a certificate through BYU Independent Study
Please note that acceptance into the certificate program does not constitute admission to Brigham Young University as a day or evening student. Should you have additional questions or need additional information, call 1-800-914-8931 or (801) 422-5078.

Certification and Accreditation as a Professional Genealogist

The certificate program should not be confused with certification. The certificate program can help prepare you, but is not required, for certification or accreditation, two separate, private forms of qualifying professional genealogists. For information regarding certification or accreditation, you may:
Write to the Board for Certification of Genealogists, P.O. Box 14291, Washington, DC 20044, or to ICAPGen, P.O. Box 1144, Salt Lake City, UT 84110. Call 1-888-463-6842 or check online at www.infouga.org. Email www.infouga.org for information regarding accreditation.

http://ce.byu.edu/is/site/programs/famhistcert.cfm