Can I download an entirely new iTunes folder directly to an external hard drive?

Unfortunately my hard drive to my last pc laptop took a dive a few months ago. Lost everything not backed up. Since then I realized we really don't need a computer anymore since my entire family only uses our iDevices (aside from my hubby's "work only" computer). So other than a few school reports, random web surfing and to back up our iDevices to iTunes I purchased a little Hp Stream 11 for a mere 200.00. Perfect for us. Well not realizing that the HP Stream's 32GB hard drive will not get me anywhere with 3 - 64gb iPhones, an iPad 1, an iPad mini, an iPod touch and an iPod touch 5.

So to reserve as much space on the laptop as I can, before I download any part of itunes (and obviously before I back up our iDevices, can I get an 1 or 2 Tb external hard drive and directly put all of iTunes right on it? I don't even want iTunes on my pc, can the whole entire iTunes program/library/folder all go directly onto an external hard drive? If so how would I go about directly doing that? All other questions regarding this are about transferring from pc to an external hd, but I want to start the whole iTunes onto the external from scratch. Any ideas if this could or would work? Thanks!

btw, our Wifi is a horribly slow DSL that bounces in and out, and it is the only thing available where I live. I tried to back up to icloud and honestly it was a huge fail. Not sure I even want our backups on it anyway 🙂