In most states licensure is not required, simply passing one of the National Certifying exams is all that is required. In those states most people attend a CLS/MLS program that can be completed as part of your BS degree in four years, usually in a 3+1 or 2+2 format where you take your prerequisites in your earlier years and complete your clinical training in the latter one or two. Florida and California require state licensure as do a few other states where 4+1 programs as described above are more common. Licensing remains less the rule than the exception across most of the U.S.
If you are thinking of going to Med School eventually, a BS in medical/clinical laboratory science is a great undergrad major as it will hit your usual med school pre-reqs, give you plenty of healthcare experience and a huge knowledge advantage of disease states and treatments over the typical pre-med student. My daughter is considering med school as one possible grad/prof degree and a BS of this type is my solid recommendation to her is she goes that way, plus it is a nice undergrad level professional degree you can get good full or part time work with during or before grad or med school.