Here is some insights I have learned over the years, especially from good priests.
First, the reader's questions: I would like to hear more about the mechanics of prayer intentions. Should I write them down in a notebook? Should I make an explicit list? How do I choose an intention for Mass? How do I choose an intention for the Rosary? How often should I explicitly pray for certain intentions, and in what form? This INTJ wants... well, a system. ;-)
First of all, let me start with serious intentions. For example, I know a family which has Masonry in the past for several generations. This type of bondage is serious and takes much prayer. She was told by priests to have Masses said, especially TLMs. And, she has made the Thirty Day Mass Novena for this intention. However, she must prayer for years against these negative influences, and she must, as she does, do mortifications, penances.
Second, TLMs are more powerful for intentions than the NO. See my post link to Fr. Chad Ripperger on this.
Third, going from serious to less serious, read my posts on "code breakers". Some people are called to great intercessory prayer in their families to break negative codes. This type of prayer may continue throughout one's entire life. Masses, rosaries, novenas and so on are helpful.
Fourth, I pray the same thing on the Rosary daily. For example, I pray for all the fallen away Catholics in my family, I pray for fallen away friends, I pray for specific seminarians, I pray for the Third Order to which I belong and for two people to find their vocations. I also pray every fifth decade for the souls in purgatory, or one special one.
For example, I am praying for those people in purgatory on the fifth decade now in the area where I am currently staying. I pray specifically for those in the ........Catholic Cemetery who may still be in purgatory.
If someone asks me to say a decade for them, that is extra, or if I offer a decade, that could be extra.
Fifth, BE SPECIFIC in all intentions. Again, I have just finished several months of novenas for a specific intention.
Sixth, yes, write things down. If I am praying for someone's healing, for example, I write that down in a list in a little book I use. (I need a new one if someone wants to buy me a new moleskin).
Lists are good. I also make lists for my own intentions.
Seven, one should pray daily for a happy death. One must love one's self enough to pray for final perseverance. Recently, I asked a priest to say a TLM for me for this intention. Why not? My salvation is not assured until I die.
I have more time than most to pray as my day is writing and praying. I believe I am called to this. Sometimes, I have prayed for hours for one person who is in great need.
Eight, one should pray for the removal of one's predominant faults, daily. This can be a small prayer.
Nine, you have children. Pray for them and over them daily. I use to bless my son and pray over him frequently. I wish I had done it more often. Pray for your children and also dedicate them to Mary. Have them do the Consecration of Mary via St. Louis de Montfort with you. I did this with my son when he was ten.
Ten, ask others to join you in prayer for hard things. For example, I have several friends praying for another friend of mine who is experiencing great difficulties in her life. Again, be specific.
Eleven, some people take intercessions to Adoration. I use to do that. Now, I just sit and look at Jesus. I have learned to do what Blessed Mother Teresa said to do-just look at Him and let Him look at you. But, if you want to bring hard situations to Adoration, I know many people who do this, especially if they go more than once a week.
I have not exhausted this subject, but I hope this helps. By the way, point Twelve is this.
Twelve, THANK GOD for answering your prayers, even if you cannot always see the answers. And, remember, the fact that you are praying for someone is a sign of God wanting you to do that-yes, there is both Providence and Predestination in our continual prayers.
One more point. I am beginning to realize just how powerful Mary is--Pray to her under the name of Virgo Potens, and pray to your patron saints. I have a list of chosen patrons to whom I pray for various things or people.
I guess that is number Thirteen!
The Mass is, of course, the most perfect form of prayer.