I currently go to a north eastern secular private high school. We play both private and public schools, mostly based on region. County lines are also very important. We are on the edge of a county but tend to only play schools in that county conference. We also play a few out of season games against schools not in our conferences but that are a good match for us (or old rivals). There are also private school conferences, which often only consist of a tournament, that we participate in. We usually participate in both the public and private conferences. This could differ depending on the school (my private middle school did things differently) but that's how most private schools seem to do things here. I'd suggest looking up some Seattle conferences for some supplementary information.
As for what sports we participate in, it really depends on facilities. My school has a swimming team because we have a pool, but some schools will go off campus to practice and have home meets (my middle school did this for cross country). We play most standard sports, but also have a few rarer ones. For example we have a fencing and golf team (and we used to have skiing). Tennis and lacrosse are pretty big where I go. It really depends on the size of the school. A lot of private schools provide a large number of different teams for a relatively small student body. Because we have such small numbers of students, my school (and many others in the area) have no cut policies for all teams. Anyone can join, but varsity status and play time is based on skill. A lot of these schools have sports programs fed into by middle schools (my school contains both a middle and high school, but even with a lone high school many people who attend come from private middle schools) where playing a sport each season is often mandatory. Each school tends to have a few good teams but overall small numbers seem to hinder most private schools in excelling at a large number of sports (too small of a talent pool). Many schools also don't stress athletics as much as academics (though that really depends on the culture of the school).
One thing you should be mindful of is that (at least in the north east) there will always be the same number of girls teams as boys teams. Some sports are called co-ed (even if girls effectively don't get to play or don't join anyways (but this can be true for boys to (ex. when I played volleyball, most meets were for girls only) to maintain this rule (though co-ed type teams are mostly in middle school). Sometimes boys and girls teams will practice together depending on the sport to some extent (ex. track and fencing) or not at all. Another thing to note is that many private high schools (yet again, I'm speaking from an east coast perspective) don't have cheerleaders. It really depends on the school. From a social perspective, being a star athlete is not a huge deal (but I go to a more academically minded school). This is especially true if the sport you excel at is not the most popular or the team is generally bad. We have nationally and internationally rated fencers and swimmers at my school (we even sent a soccer player to the World Cup in South Africa in the lower age tournament), but while people are aware of these accomplishments it doesn't really ensure any popularity. But that factor will all depend on the type of school you have and it's culture. My school is academically oriented with focus on the arts as well as athletics, which take a back seat to general learning. Sports are often seen as a great way to build a college application, enticing people to stay with a sport they don't really care about and don't put much effort into.
Sorry for the long and repetitive post. If you have any other questions about private school sports, or private schools in general, please feel free to message me.