There are quite afe logging applications. Most of them are pretty feature rich, many look like they were designed in 1983. If you don't need a load of add ons the Cloudlog is a great option. I run mine on my own server that is just a Raspberry Pi and small SSD drive. The secret is in the software....And it is all free!.
Where to start?
My experience is based on Diet Pi, a really lightweight Raspberry Pi districution with a load of useful software that can be loaded with a few clicks. I had originally used a RPi Zero W and Jessie Lite (which worked nicely btw) but it needed a bit of faffing about to get it to work. I guess Diet Pi will work with a RPi Zero W but you'll miss out on the many features. In fact you don't really need to limit yourself to a RPi. Any SBC that can run DietPi will do, there is a list of sorts here.
Ok, so I'm going to make a couple of assumptions.
1. You've got your RPi and know how to ssh into it.
2. You've downloaded and burnt the image to your micro SD card
3. It is powered up and ready to go. By that I mean you're in PuTTY or your fave ssh app
4. You have got yourself a copy of an ftp app, like Filezilla and you have a rough idea how it works
Instruction #1 - launch the software utility