Learning — memorizing — common Catholic prayers is important for two reasons.
First, it is a connection with Catholics and the Church throughout the world. Whenever someone begins one of these prayers, you will be able to easily join in with everyone else. That is shared or community prayer. Jesus Himself has suggested that we pray together: "For where two or three are gathered together in my name, I am there in their midst.” [Matthew 18:20; NCB: New Catholic Bible]
Second, these are wonderful prayers. These are prayers we can pray in our personal lives. Sometimes, if we might seem not to know what to say when we pray, we can always pray one or more of these (or other) prayers. But praying from the heart in our own words is also very important to do. That, especially, helps us to get closer to God... to Jesus, Our Lady, and the saints.
We will put these prayers into three categories:
(1) Regular prayers for use in our personal prayers and also when praying with other Catholics.
(2) Prayers of the Mass. Yes, these prayers are in the missals in the pews. But that's not good enough. You need to know these from memory. These, too, are good and rich prayers. (One or more might also be in the first category just above.)
(3) A number of the Mass prayers in Latin. The church expects the Faithful in the pews throughout the world to know certain Mass prayers in Latin. This is not some old thing, perhaps before the Council of Vatican II, which has fallen away or is even discouraged by the Church. This is a current expectation as found in current Church documents today (see the footnote below). It has been reiterated by more than one recent pope. Sadly, except for a relatively few priests, we rarely hear those prayers at Mass in Latin. Most of the Faithful no longer know them. Some clergy even wrongly take a strong position against any Latin at Mass, regardless of the official position by the Church itself.*
In spite of that, it is not likely that we will have the time to address this category of prayer. We will concentrate only on the first two categories here.
The word "memorization" has fallen out of favor in recent years. But that is precisely the correct word for learning prayers. One can not learn many things in the world, including shared prayer, without memorizing, practicing, and actually praying them.
LEARNING CATHOLIC PRAYERS
The Sign of The Cross
If you had to guess, you might think that we would start with the Our Father (The Lord's Prayer). But that's not where we start. Our first prayer is actually the Sign of the Cross. Many more prayers will be coming. Be sure to completely learn (memorize) and practice them.
"In the name of the Father, and of the Son, and of the Holy Spirit. Amen."
At the same time as you're saying those words, put your left hand flat against your body just below your rib cage. Then, with the fingers on your right hand, touch your forehead (the Father). Then touch your left hand which is already on your body... waiting for you (the Son). Then touch your shoulders, first the left one, then the right one (the Holy Spirit). [The Orthodox can go from right to left.] Finally, at the Amen, you can fold your hands, as if in prayer (which you are), in front of you. And that's it!
There are less formal ways many people make the Sign of the Cross.You can leave out the left hand placement, for example. Others go very quickly with the hand not even fully touching the shoulders or the forehead. While it acknowledges the Sign of the Cross, a concern is that, when rushing it, people lose the sincerity and understanding of what they are saying. Especially in some cultures, some end by making a quick cross with two fingers and kissing the finger cross. So options do exist.
---------------
Many non-Catholic Christians like to end their prayers with: "We ask this in Jesus's name. Amen," or, "In Jesus's name we pray," as examples. After all, Jesus said in Scripture that whatever we ask in His name will be granted. Therefore, why not ask for everything in His name?
But although there is nothing wrong with praying in Jesus's name (Catholics can and do), that scriptural verse is taken out of context. That's because Jesus did not literally mean for us to do that every time we address God in prayer. If you want to actually know how Jesus taught us to pray, look at the Our Father. That prayer was specifically given to us by Jesus Himself. And nowhere in that prayer or in Scripture where Jesus gives us the prayer, does Jesus tell us that we should invoke His name in it. In fact, Jesus is telling us to specifically address His Father in that prayer: "Our Father, Who art in Heaven..." [You can also look at the Chaplet of the Divine Mercy, but it's too early to look at that right now.]
Many years ago, I was speaking with a Baptist deacon. He absolutely insisted that all prayer must be directed to Jesus. When I pointed out that the Our Father is prayed directly to the Father and not through Jesus — it's easy to make that case since the prayer is so clear about it — the Baptist deacon became confused. He had no answer. A week or so later, the deacon told me he had talked to his pastor and that, apparently, the Lord's Prayer is an exception or something or other. It's been many years now and the precise details have long since been forgotten. But, as the deacon found, the case for praying absolutely everything in Jesus's name cannot be made.
Nor do we see that when we carefully look through Scripture. Jesus didn't even mean it literally when He told us that whatever we ask in His name would be granted to us. After all, in spite of every prayer and every request being in Jesus's name by non-Catholic Christians, I can pretty much guarantee that they are not automatically getting everything they ask for when they pray in Jesus's name — even though, they believe, Jesus Himself said that they would.
Well, what actually did Jesus mean then? He didn't mean that if we just throw out rote words using Jesus's name after every prayer because we're told to, that all of our prayers will be granted. He meant that, if we pray from the heart with true sincerity and love for something of value, that He would grant it. What does "from the heart" mean? It means we pray in the language of God. Since God is love [see 1 John 4:8, among others], it is suggested that God's language itself ... is love. We should be praying to God with love.
Most people pray with their words only. We need to learn to pray in and with love, the language of love, because that is how God speaks to us, it is what God is. That takes practice. In the end, our sincere prayers, spoken in love, become different from the rote prayers of the past where we just unthinkingly added the words of Jesus with nothing actually coming from the heart, coming from love.
----------------------
Our Lady (Mary) has said that the Mass is the best prayer. If you look carefully at the prayers at Mass, only a very small number are addressed directly to Jesus. The vast majority are specifically addressed to God the Father.