Let’s break down the Hail Mary for a starter, it is a bit cheesy and cliche, but why not?

(you can and still should do spontaneous prayer btw. Early Christian prayer was both memorized and spontaneous)
“Hail Mary”
Hail= formal way of saying “hi”
Hail Mary= the way Gabriel greeted Mary at the Annunciation
Hail Mary also means we are remembering what it was like when it was first announced to another human being (Mary) that God would become man
Full of grace= this is the basis for the concept of the Immaculate Conception. Mary is the New Eve, she fulfills the proto-evangelion in Genesis, she will help crush the head of Satan. God has filled Mary with grace and we are celebrating that fact
The Lord is with thee= this is quite literal. The Lord is literally with her, literally in her. He’s also there in a more abstract to strengthen her as she will give birth to Christ. It also kind of celebrates the fact that God chooses the most unexpected person as a tabernacle, a poor woman but one with many virtues we should probably strive to imitate. If you really want to stretch it, “the Lord is with thee” also kind of applies to us as children of Mary and as people who have also received Christ in a different way and can rejoice at the incarnation. It is supposed to motivate us.
blessed art thou among women= This is what Elizabeth told Mary at the Visitation. Mary is blessed (in part) because she gets to hold Christ. The way Elizabeth greeted her mirrors Israel’s celebration around the Ark. The point is that the whole world rejoices because of the Incarnation of which Mary is the carrier and thus she is blessed among women, she gets to have a special way of enjoying a blessing we all get to rejoice at
and blessed is the fruit of thine womb= This flows from the last point. It explains why Mary is called blessed, because of the fruit of her womb
Holy Mary, mother of God= Mary is sinless (through the grace of God) which makes her holy apart from the fact that she was the mother of Jesus but by calling her “mother of God” we are once again acknowledging the divinity of Jesus and the two natures, fully God and fully man
pray for us= quite simple. After we’ve stopped a moment to have a mini-celebration of the Incarnation we ask Mary to pray for us.
sinners= this is simply a recognition of a fact, we are sinners. I’d call it a semi-penitential recognition, it kinda makes you stop a moment to think about all your sins and think about how you are not like Mary and certainly not like Jesus
now and at the hour of our death= this is more of a recognition of how though death is going to be like for us and how we want to stay strong at that time. It’s a built-in petition.
You don’t have to say rote prayers. Any Christian tradition that is even slightly old uses them, not using them is a modern thing, but they are not mandatory. You are encouraged to use your own words, it’s a great thing to do if rote prayer does not really appeal to you! I use my own words all the time and I’m sure almost everybody in this thread does.