I've seen church and school websites done all kinds of ways. Maybe there's no one best solution, but when I worked with Calvary Lutheran in Murrysville, PA, we decided to try to revolutionize the combined church/school website. Maybe we did just that. I'm going to describe the problem and the solution so that you can see if this might work for your church and school.
One Site Problem
The one site problem is that either the church or school seems more important. Either that, or it ends up being a landing page without much of a purpose. Another problem is in the main menu, since it can be overwhelming as a single website. When that menu sticks around for other pages, navigation ends up being awkward. If you're on the school side of the site, you don't really need to see all the church menu items, but a sigle website kind of forces this hand. I think the result is that larger websites tend to use a landing page scenario that leads to church home and school home sites. Smaller websites might combine the church and school site, since the menu is more limited. Logins and permissions can also be a challenge in the one-site solution. Not all websites allow certain people to only edit certain content. Schools that are tied in closely to the church might be able to use the single-site approach, but those schools with limited church family enrollment will likely want their own page of sorts.
Two Site Problem
Another solution I've seen is two totally different sites. The church gets its own site and url, and the school does the same. But it often doesn't really do the same. Often, these sites don't look alike, since a volunteer at the church made one while a marketing firm for the school put the other one together as part of a recruiting effort. Even if they do look alike, people might wonder why two pages exist. It's probably harder for church and school folks to find what they're looking for, especially if it's buried in an old article somewhere. There's also the problem with the perception that the church and school are not related. The problem is that both ARE related, so now you'll need logins for the principal on both sites, as well as for pastors and secretaries. For very large school websites, this might make sense, but most Lutheran church and schools don't really want or need an extra website.
The Three Site Solution
When I was teaching, I liked to use multiple-choice tests. They took three times as long to create, but I saved all kinds of time every single time I used them. Some of those tests were used for over a decade, teaching three sections of the same course. That's a lot of efficiency from a little extra time. When I came up with the three-site solution for church and school websites, I tried to come up with a way that would work the best down the road, even if it took a little more time from the start.
The three site church and school solution was done in Joomla, but I am sure it can be done with another CMS, too. Basically, I needed to create a school website and a church website, complete with separate menus and templates (themes). After that was done, I needed to add a third website that was just for the home page, but because it was really a third website, it's much more than just a landing page. The cool part is that the websites are all separate entities, even though they are all the same site. Almost a metaphor for the Trinity. Same database and users. The fact that the templates are technically different means I can reverse the colors in the church from the school: a subtle yet effective separation.
The challenge of the three site system is that it took me the same amount of time as if I was actually building three websites. That's kind of a hard sell to an organization, since it works so well as a single website. I can't exactly charge for three websites, so I end up working harder, which means I'm probably the only website builder around who's going to offer this kind of website.
When you try something no one else is willing to try, there's will be some trade-offs. While I do believe this is the best way to create a church and school website for most LCMS church/schools, some comprimises had to be made in order to make the site work. Check it out for yourself, keeping in mind that the website takes the place of the other options above.