So,how do I download and update my iPhone 5s to iOS 12.5.5?

There are several ways. My preference would be to do so via a computer with Finder (Mac) or iTunes (PC). It's the best way to do it from a technical standpoint. Every update is a complete iOS build and it requires less free space on the device. I prefer using a computer because it requires less free space, and because it will automatically perform a backup before proceeding with the update. You never know if there might be a problem and the installation is corrupted, so a backup is extremely valuable.

Update your iPhone, iPad, or iPod touch - Apple Support

If there's not enough space available, sometimes the only way would be to back up manually, restore the device to factory settings (which would then install the latest version of iOS for your device), then restore from the backup to the device.

But you could also update via Settings > General > Software Update and see what updates it has. You'll have no control over what version it gives to you, although if you've got a really old version it should just give you the option of a full installation of 12.5.5 and not just a series of incremental versions. It will also only download the update through Wi-Fi, and not cellular data. That is supposed to be the easiest way, but will require quite a bit of available free space on the device to download and unpack it.

There are also automatic updates, but many prefer when to control when updated install.

What version of iOS is it running now?