As long you buy the cellular-capable Apple Watch, then you'd be able to do most of the same things you can do when the iPhone is nearby. The GPS-only model requires the iPhone in order to get connected to the internet (although it can also use known Wi-Fi access points that as long as they're not "captive" Wi-Fi networks — ones that require you to go through a sign-on page).
However, most of the health monitoring features don't actually require an internet connection, so if it's primarily for that you'd be okay with either model.
Apple actually has a pretty good support article that lays out what the Apple Watch can do when it's on its own:
Use your Apple Watch without your iPhone nearby.
You'd still need to use the iPhone to actually check the health data, since there's no equivalent "Health" app on the Apple Watch — all of that data gets recorded into the health app on the paired iPhone (this is likely the main reason these features aren't available on a "Family Setup" Apple Watch — there's nowhere to record anything).
You'd also need to use the iPhone for things like installing watchOS software updates.