The official supported way to get your photos from Flickr is using their "Request My Flickr Data" button on your flickr account page. This button will not get you your data immediately, but will eventually guide you to download a file containing all of your photos and much other Flickr account data too:
The download also includes "Flickrmail" sent and received, testimonials, and comments on sets among other data. All the data other than the photo files is in JSON files (at least one JSON file per photo plus the others miscellaneous files.
At this point you technically have all your data but it is hard to do anything with. At least you have a local backup.
There's an article by TechAdvisor that's still pretty accurate about the pros and cons of some other services, and it does describe how to get your photos elsewhere but it involves the download-to-local process described above.
This solution gets all your photos, photo metadata and account metadata into your own hands on your own device. The metadata is in a Flickr-specific format in JSON files.
This solution is a complete backup but not a great move solution.
The source code (markdown) for this article can be found on GitHub.