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
You should end up with this screen....
Here is the magic DietPi software tool. Focus now! Don't get carried away and install everything. Tempting as it might seem. Now just as a note my installation might be different to yours because I have my own domain that is served by the RPi and that uses No-IP. Also available through Diet-Pi. Classy eh? Perhaps I should do somethingon that as it wasn't a first hit winner. Anyway back to the job in hand. Select the right install for you. I went for the nginx sqlite and maria db options (not too light but light enough to avoid bogging the RPi down)
Instruction #2 - Make a cup of tea
Just wait for it to finish and sip that nice warm cup of tea. Coffee is ok but lets face it tea is way better.
Instruction #3 - Download the cloudlog files and folders from GitHub, unzip 'em and upload them to the \var\www folder.
Thats the hard work done.
It is now the simpler task of following the instructions on the Cloudlog wiki to get your account(s) set up and you're off and running.
If you want to take it further then there is the opportunity to add CAT control if you need that. I just type in the details for now, for the amount of QSO's I make then it is no big deal but if you're a 10,000 qso per year kind of ham then it might get a bit annoying. Still there is an answer for that.
Enjoy. If you have any feedback then get in touch and let me know how you get on.