I'd say, check out the public schools in your area. Your taxes are already paying for them, and (as a product of a private school, grades 2-8) private school can be hell. I don't know where everyone else found wonderful experiences at private school, but at my school my parents were the wrong denomination, wrong political leaning, too poor, and *gasp* my mother worked. Together with my being rather . . . quick . . . intellectually and sensitive socially, these factors made my life at private school absolute *hell*, and it wasn't just students. It was teachers, principals, parents, supporting and adding to the student bullying/general asshattery.
The minute I entered public school, I was a different person. The classes challenged me (they put me in GT/Pre-AP/AP) there were people like me there, and those that weren't like me didn't care that much. Not enough to torture me, at least, and that made all the difference. However, we DID move from one part of town to another so I could go to a very good public school - that research took a while, but it was worth it. We did research on the private school I went too also, but there was no hint of the evilness of the place when we visited - it only emerged when the $$$ was in hand and I was in the classroom.
Another quite note about private v. public schools: in private schools, your kid's going to get a very different idea of society than in public schools. Even in the good public school my boyfriend teaches at, there are students from poor families. In private schools, your child likely won't know many non-well-to-do children at all. And, let me tell you, when your child goes into the real world, there will be people who have trouble feeding their families around (though I wish that were different). That's not when you want your child to realize that not everyone has parents with PhD's and MD's - I had a friend like that in college, and he *gasped* at hearing the amount of money my fairly well-off family "made do" with. You don't want your child socialized with just a small fragment of society, or the rest of it is going to be a real nasty surprise.
Homeschooling: It's likely, as a doctor and a grad student, you won't have the time to home-school your kid, and it's not optimal. Teachers and parents each have their roles in the development of a child, and when you combine them into one being, the kid doesn't have the option of running to teacher/parent when having problems with their counterpart.
Then there's the fact that, if you have a gifted/talented (that's what we call it in Texas) child, educating them is going to be an ordeal. GT kids are different, and way harder to teach than non-GT (or simply very smart) children. The fact is, teachers are certified for a reason, and GT certification has higher requirements for a reason: not everyone can teach, and not everyone wants to teach. When it comes to truly GT kids, the difficulty goes up quite a bit - they're fast and intuitive in some areas, maybe all areas, but when they don't want to do something, they can't be bothered by love, money, or death to do it. I should know - I was one, and I made some certified teachers want to murder me. It takes training and creativity to learn to deal with the quirks of a truly smart or GT student - so, unless you feel up to relating all your lessons to horses (I, like most GT kids, fixated on one subject to the exclusion of pretty much everything else) for 5+ years . . . homeschooling a GT kid is not for you. TBH, I still have a horse fixation, and the only reason I started D&D was because I could play someone with a pet horse. The video games I'm interested in? Most involve horses. That's *very* typical GT-ness.
