Part of the problem is that Apple has changed the way it manages photos over the years. Photo Stream was a way to get images off your phone in near real time. However, the photos in Photo Stream are removed automatically after 30 days, so it requires that you copy them to somewhere permanent. I don't recommend using it anymore, as there are many better solutions. There were also a couple of Apple photo management apps including iPhoto and Aperture that still work, but have been deprecated and aren't being updated.
iCloud Photos is the current Apple photo management feature. If you turn it on all photos you take will be copied to iCloud in near real time, and also copied to all devices that have iCloud Photos enabled on them, including other iOS devices, Macs, and even Windows (with the right app installed). They can also be viewed by logging in to iCloud.com with a browser. It has 2 downsides: The photos are replicated on all devices and synced, so if you delete a photo from your phone it will also be deleted from your Mac or other devices. And they take up space on all devices. The other is that if you want to manage them in some other app (such as Lightroom) you must export them using the Photos app on your Mac. If you have iCloud Photos enabled on your phone then your photos will not be included in an iCloud or iTunes backup. There's more details here: Keep your photos safely stored and up to date on all your devices - Apple Support
You don't have to use iCloud Photos. If you turn it off you can also treat the iPhone as a digital camera, and copy your photos to your computer using a USB connection. This doesn't use iTunes; if you plug the phone in it will be recognized by your computer (Mac or Windows) as a digital camera. You can use any app that you would use with a digital camera to import photos. On a Mac the built-in tool for this is the Image Capture app, but within Image Capture you can set it to connect to any other photo management app. See: Import photos and videos from your iPhone, iPad, or iPod touch - Apple Support and What to expect when you import edited photos from your iPhone, iPad, or iPod to your Mac or PC - Apple Support. Note that if you use this approach you will lose any edits that you have made to the images on your phone, including any filters that you have enabled. The metadata is saved in a separate "sidecar" file that is not imported.
Backups are different. The iCloud and iTunes backup feature backs up the phone's content that is not synced any other way. For example, it will back up your contacts if you do not have iCloud Contacts enabled, but it will not if you do. Music and apps are never included in backups, because they can be reinstalled from the iTunes media library on your computer. For more information on backups see: About backups in iCloud and iTunes - Apple Support
Syncing is mostly one way, from iTunes on your computer to your device. So it isn't truly syncing. See: Sync your iPhone, iPad, or iPod touch with iTunes on your computer using USB - Apple Support
To create photo albums on your computer that include photos not taken on the iPhone you use iTunes to sync the photos to your phone. With the phone connected, look at the Photos tab in iTunes. See the 2nd major topic in Keep your photos safely stored and up to date on all your devices - Apple Support(this link is also above in the iCloud comments). This approach is a little kludgy, because there's no way to remove the photos from your iPhone except by connecting to iTunes and "unsyncing" them. I use 3rd party tools instead; either Google Photos (which is really amazing), or upthere.com, which is a startup created by former Apple employees. Both of these keep the photos in their respective clouds, and only download them when you want to view them.
This should get you started. Please feel free to ask about anything else that occurs to you.