For instance, if I use a VPN at a local high school, can the internet administrators at the school tell I am browsing Youtube?
If you've configured properly VPN and you don't use their DNS servers but use some other thru VPN as well you should be safe. I'm assuming that you own the computer you're going to use. Otherwise other things like GPO, lack of administrative rights etc may get in your way when you even try to setup VPN. And even if it would allow this I wouldn't trust that anyways and would opt to use your own computer with VPN.
As long as you are in control what's on the PC (as in the administrator didn't install any "monitoring" software on it) and you connect to a VPN service which changes your DNS also (like PingBuster) then they cannot see what you are doing.
All they see through sniffing are encrypted packets. As you don't use their DNS server either they cannot see the DNS requests. So, this way you are 100% safe. The only thing they see is to which VPN server you are connected (as the packets will need to be send to somewhere obviously).
For one, most school network administrators have access to client-side software that's going to get in the way. Such software is designed to limit your browsing experience on the actual computer you're using, by monitoring your activities even if you're routing through a VPN, or by limiting the sites that you can go do through settings and security (For example, those IT administrators adamanantly opposed to YouTube could install software which prevents flash from running on the YouTube domain).
So, you may or may not be able to access YouTube through a VPN, and even if you are, you may have your activities being monitored. Network IT administrators tend to take this stuff really seriously too (Which makes sense as it is their job). So, prepare to handle the wrath of your IT staff if you take the risk :P
When you use a VPN it'll hide your ip address where you can not be traced and it will not show on a Admin's computer that you are even looking at certain sites. VPN's are bascially just to get you past blocked content without being detected. Onavo is what I use to use and it works really good.