There are several strategies for aiding the memorization process. While many singers just sing through their songs over and over again until they know them, and this certainly gets results, it is not the most efficient way.
I suggest you not even think about memorization until you have mastered the musical elements of the piece, so that you can sing the pitches and rhythms accurately and expressively, with good breaths and appropriate dynamics. Once that is secure, focus on the text. Our brains attach musical pitches and rhythms to words easily, so once the words are memorized, it is very likely the pitches and rhythms will come along with them (provided you learned the pitches/rhythms thoroughly first).
The first step is to take out your music and copy down the text in your own handwriting. Doing this with paper and pencil is important, as it has been shown in countless studies that physical writing leads to greater retention than typing. When you write you engage yourself kinesthetically (physically) by moving the pencil, visually by seeing the words form from your hand, and aurally because it is impossible for humans to read and write words without hearing them inside our heads. This is the process of ‘audiation’ that I often bring up in discussing solfege and music reading.
Use your own handwritten copy to refer to as you rewrite the text as least 3 times.
The second step will seem odd but this is where you will start to really make progress memorizing. Take your handwritten copy of the text and on a new sheet of paper copy down just the first word or two of each phrase or sentence of the text. Once you’ve finished copying only the first word of each phrase, flip over the full text and try to fill in each phrase by memory. This should be harder than step one but much easier than remembering the full text on your own. More often than not, it is the first word of a new phrase that trips us up, and once we get that word we can get the entire phrase with little to no problem. This is a trait singers share with actors, as you’ll often find forgetful actors saying “line” and once they get the cue of the first word, they will rattle off a monologue of several pages.
Now that you’ve seen the power those first words of a phrase hold, again copy down just the first words of each phrase. Now repeat just these first words out loud, in order. Do this several times, until you can easily and effortlessly recite the first words of each phrase of your piece one after the other.
Congratulations! Now that you’ve memorized the start of each phrase, you have likely memorized the entire text. Grab a blank sheet of paper and test yourself. If there are still areas of difficulty return to reading aloud to polish those sections.
You are now ready to return to the music and attempt to sing the song from memory. I caution against going to this step too early, as many singers rely on the music to remind them of the text. If you don’t learn the text separate from the music, it is likely that with the butterflies in your stomach and the stage lights on your face you will blank on one or more words throughout your performance. Put the time in to learn your text and you will feel incredibly confident going forward.