This list is not even close to comprehensive! note: some of these careers don't require postdoc training or even graduate school
A majority of scientists go on to jobs as:
MDs and other medical professions incl. PAs, nursing, and technicians including neuromonitoring (surgical neurophysiologist), anesthesia tech, med tech, scrub tech etc.. and also research and clinical trial technicians
Pharmaceutical research or sales and service
Staff at NIH/NSF/ other government institute or corporation
Journal editor
Science writing for non-scientists
Science illustration
Other journal or foundation role
Non-profit and/or public health services
Museum work and/or art, display, and curatorial fields
Fisheries/ wildlife management or other field position
Patent or other law fields
Consulting (statistics or expertise in a field (genetics etc…))
Entrepreneur or small business/startup (maybe something you developed in grad or postdoc) Check out Mekentosj
Technology design
Analysis skills used in alternative fields
Military
Advertising
Grant writer/support --positions at most Universities focus on aiding faculty write winning grants (when a school gets 60% of the amount of your grant; it's very important that they help you write and obtain grants—grantsmanship is a skill)
Professor, research associate, lecturer, instructor, administrator etc. in academia
Many more, and this list is in no particular order...
Research or Teaching Professor in Academia
Large R1 Universities and Medical Schools
Intramural NIH laboratories
Private Institutes
Small Primarily Undergraduate Institutions (PUIs)
Primary or Secondary Schools—requires additional training
Grad school pays you (~$24,000/year plus benefits)
Often 1st year paid as a teaching assistant (TA) and subsequent years may be RA (funded by PIs grant or pre-doc grant)
It’s a major commitment, may want to take a year or two and figure it out (be a postbac, do a tech job or do something very different—experience the world!)
Make sure a PhD is what you really want (see career options below), what about a Masters (you can always find a Masters program and switch to a PhD if you are enjoying it).
Harvard and other big name programs may have 300+ applicants for <30 spots
Conversely, state universities may accept all that apply (assuming they are qualified) and often have excellent grad programs (and certainly have great faculty)
Taking the GREs
Take a GRE course, you'll learn to take this type of test and improve your score
It's not the only thing that gets you in (undergrad research experiences, publications, and other life-rounding experiences are prized)
How to choose a school/program/lab
What kinds of articles have you read that excite you? (this may provide a good indication of what type of program you should pursue)
Are there particular research areas (maybe from your courses or invited seminar speakers) that you are interested in?
Do you want a broad initial training (rotating in the first year) or to start directly in a lab?
Some programs (e.g., some organismal biology programs and others) require you to find a lab in order to "get into" graduate school—this is very different than molecular biology programs, for example).