There is no 'correct' way to learn a new piece. There are many good ways, but they are quite subjective. They will depend to a degree initially on how good one is at sight-reading. In fact, several of the guys I have played with over the years are so good at sight-reading, they never have to actually 'learn' pieces. If that's not a great indictment to learn to be a good sight-reader, I don't know what is!
Back to the question in hand. Some will get pretty good on each hand separately before putting them together. Some will work, more slowly, on getting it together from the start. It does depend very much on the sort of piece: with a rhythmical left hand accompaniment, the two together will be a good idea, whereas if it's more of a question-answer type, separately works well.
There is also the important factor, missed by so many learners, that it's the tricky bits that need to be practised, not the 10 bars that you play well before and maybe after them. So isolate those nasty bits, and just work on them, otherwise a lot of time is being wasted. Once a section is learned, there is no special reason to keep playing it, but rather wait till the part before or after is as good, then run them together.
Some need to see (and hear!) the harmonic structure and format of the piece, so listening to renditions first is good for them. Some like to do it in little bites, say 2 bars at a time, and again it depends on the way the music goes - if 4 bars makes a sentence, then 4 bars is a good bite. Some like to start at the beginning (a very good place to start, so we're told), but others will take a passage half way through, or even the last part, so when it's nearly learnt, they already know what's comong next.
Some may lock themselves away and repeat as many times as it takes, before letting themselves out,that works for me!, whereas others prefer simply a few minutes on a particular part, and return a couple of hours, whatever, later, and do the same.
Some find they're most receptive at certain times of the day or week, so attempt to put the work in then. It may be first thing in the morning, when they're freshest, or later at night when the worries of the day have subsided somewhat. Some need the high expectations of their teacher to spur them on, others work better without external pressure. Some give themselves a deadline - I'll perform this for my friends at the end of the month - while others plod happily on, knowing it'll be there in the end - if they live long enough...
In amongst all this, you need to find YOUR best way, by trying out some of the suggestions. No one way will be best for each and every piece, but those are some ideas for you. Good luck!