I'm with others who prefer to start from text (getting any story 'straight' in my head), and step 2 is to get to grips with the tune in a basic version.
I usually sing unaccompanied so I would learn the bones of a tune, often from notation, till I was secure with its shape, where it rose and where it fell -- and all the time ignoring rhythm and/or decoration (which is why I don't like learning from another singer's recording, although I know singers who listen to recordings on endless repeat on long car journeys when learning new repertoire).
And when I have the tune solidly in my head, I turn my attention to the words. I like writing the text out by hand from my source -- a bit like doing lines as a punishment in school? -- and I start fitting text and tune together, which almost certainly involves changes to suit my ear: I may take out, or add, a couple of small words to suit the pulse of the tune; I may put in a wee run of notes to accommodate a line with extra syllables etc.). And after lots of repetitions, I usually rewrite the words in skeleton form so that the key words that sit on the pulse notes of the tune are the only words visible -- and then clutch this aide memoire as I get on with the learning process.
There's no recognisable time scale for this process -- it's just till I feel comfortable with the results. And the first public outing would always be in a small group of like-minded people.

Bottom line is that there's no single Right Way -- I just happen to work best with a system which involves the physical component of writing coupled to my sense of the pulse of the words (which is seldom a strict rhythm).

By the way, I'd love to see someone doing the backwards learning -- starting from the final verse -- described above. Halfway through my 75th year, the very thought of it gave me the heebie-jeebies!