I don’t do intentions for vocal prayers (rosary, Divine Mercy chaplet, etc) for myself much at all - I figure that if He wants me to have something, He’ll give it to me. I trust Him to give me everything I need for my own sanctification. If that means an exceptionally powerful consolation during mass, so be it. If it’s an $1100 car repair bill, then that’s a gift from Him too and I am also grateful for that.
During my daily mental prayer (as St. Teresa of Avila describes it, my time to talk with one of my best friends and get to know Him better), I often pray for the grace to be able to Love him as He deserves to be loved. I ask for God to make me humble, but not to let me know that I am humble, and Holy but do not let me know that I am holy, for in knowing my humility and holiness I would quickly sink into pride and lose all the grace given to me. I ask for the grace to accept His plan for me, and for me to grow as close as I can, as fast as I can, and if that means suffering, so be it. If that means sacrifice, I trust in Him that it will be for my own good. I pray for His love often as well - St. Francis de Sales says that a person can ask nothing greater in prayer than God’s love.
But I do ask for grace for others, mostly for the sanctification of priests, for the conversion of souls, and I offer up many sufferings to God for the Holy Souls. I figure if I pray for the conversion of souls and the sanctification of priests, that covers just about everything. Make us holy, and make sure the Church survives so we can be with Him here on earth.
There’s not much more to ask for really, unless a person wants to be specific about things. But usually, if you think about a situation, it will boil down to loving God and doing His will, or conversion of souls, or sanctification of priests. For example, when my boss was giving me such a hard time, I prayed a 54-day novena for the conversion of her soul. And when she fired me a couple of weeks ago, I prayed for her soul too. I also prayed for the grace to be able to love God despite the trials going on in this physical world of mine, and for the grace to discern His will for me. I also prayed that I could find a job that would allow me to continue to go to daily mass, as it has helped me grow in love for Him by spending that time with Him every day. I also thanked him for the opportunity to show Him that I loved Him through this trial and suffering. And, truth be told, thanked him that those Lessons in Humility that my old boss was giving me on a daily basis are now gone, and now I’ve moved into that much larger Lesson in Humility called the Job Search.