Do you have any links to tutorials on getting LAMP up and running on a Pi?

There's probably a boxen script for getting a lamp stack up and running on a lot of linux boxes:

https://github.com/boxen/our-boxen

It'll save you a lot of the configuration problems initially, until you're ready to configure.

I've none that I know of personally, but a quick google shows an absolute tonne of tutorials. There's really no shortage of instruction when it comes to setting up a basic webserver.

while part of me knows that a Pi will work, I feel like minimum for a home server really should be an intel NUC mini box.

I've had a 450mHz G4 PowerMac running my local services for close to five years now. There's really not really a whole lot home server needs to be doing, CPU wise.