I had my own very successful private tutoring business for 8 years.
The tutor in the family sounds overqualified for elementary school kids. If s/he wants to go down that route out of personal preference, by all means, but it's a completely different skill set working with younger children. It's more about relating to them and presenting the information in a fun interactive way then it is in knowing anything specific. It also pays less. On the science side, I think the fat part of the demand curve is junior/senior level high school classes (chemistry/physics) and entry-level college classes (core requirements or people who thought they were going to be a chemistry major until halfway through their first semester and now just need to pass desperately before switching to psychology).
If the prospective tutor can help with math effectively, definitely look into that. There is absolutely never a shortage of demand for math tutors. Additionally, clients will stay with you longer. People can "get by" earth sciences or biology and maybe need some help with chemistry or physics, but a lot of people need/want help with math starting with algebra I, all the way through SAT/ACTs and graduation.
Your #1 advertising medium is word of mouth. You can start out advertising on Wyzant and/or Craigslist, etc, but most families will take the recommendations of their friends. Get a few clients, even if you have to start with lower rates ($30-40 a session) and once the referrals start rolling in then up the rates on the new clients. I tried to (almost) never raise rates on existing clients, but just on the new ones. I always felt bad essentially saying to someone "Thanks to all your referrals, I am now so busy that I'm raising your rates!!" I might as well throw in a sinister laugh and twirl my mustache at that point.
There will be turnover. As long as your business in growing overall, don't obsess over the few people who quit after a few sessions. The old adage "the best way to guarantee failure is to try to please everyone" holds very true. You can't be everything to everyone, so as you progress you will learn your strengths and weaknesses and should accept the market you appeal to, whether it's biology or math or chemistry, whether it's homework help or SAT prep, whether it's high school kids or elementary school kids.
Working as a volunteer tutor is great. It will help determine if you really want to do it. It will also give you an appreciation of how much better you are treated when someone is willing to pay money for your services (I learned that people don't appreciate what they don't pay for through my years as a volunteer tutor, as much as I loved my time doing it overall).
For location, I would always just say "whatever works best for you." As a result, 90% of my sessions were done in the clients' homes, and 10% in the local library.
Finally, just remember that tutoring is a service-industry job. As you move up the pay scale, you will be just one of a parade of servants coming through your wealthy clients' homes: the gardener, the pool guy, the music teacher, the college admissions consultant, the plumber, and then you're there for your hour as well. You will have to be deferential much of the time, and always on your clients' schedules: never cancel a session and almost never move a session, even if it's extraordinarily convenient for you. Never be late, but also, never be more than a couple minutes early, even if it means sitting in your car for a little while. "Yes" should be your most-oft used word and you should always be smiling, even if your favorite uncle just died. You exist to make their lives easier, and the more you can do that the more you will be liked, the more business you will get, and the higher rates you can charge. I maxed out at $100/hr in a MCOL area. In places like NYC or San Fran some of the best can get multiples of that.
Just be warned--the people who pay higher rates also have considerably higher expectations. There's nothing wrong with sticking it out at a certain level (say $50-60/hr) to lessen the demand and stress.