In the context of your question, "praying for" refers to supplication. As a rule, it is preferable, yet permissible, that people pray for themselves than to ask others to pray for them. When asked, though, what you said you are currently doing — make du'a' to bring forth good things to them, and to avert bad things — matches the level required.
Typically, I would tell them that I would pray for them, then do a supplication afterwards when I am by myself, not necessarily right there at the time in front of them.
It is quite normal that one feels that there must be something more to it than that, but there is not really. A man, who had embraced Islam, approached the Prophet ﷺ about the same topic of your question. He asked the Prophet ﷺ about the form of supplication that he should use, i.e., how to bring it up to the level of the supplication of the Prophet ﷺ and that of Muadh ibn Jabal, a major companion. The Prophet ﷺ asked him what he supplicated for, and in spite of the simple form of the man's supplication, the Prophet ﷺ endorsed it, and said that it was basically what we all go around [using more articulate words] to attain:
حَدَّثَنَا عُثْمَانُ بْنُ أَبِي شَيْبَةَ، حَدَّثَنَا حُسَيْنُ بْنُ عَلِيٍّ، عَنْ زَائِدَةَ، عَنْ سُلَيْمَانَ، عَنْ أَبِي صَالِحٍ، عَنْ بَعْضِ، أَصْحَابِ النَّبِيِّ صلى الله عليه وسلم قَالَ قَالَ النَّبِيُّ صلى الله عليه وسلم لِرَجُلٍ " كَيْفَ تَقُولُ فِي الصَّلاَةِ " . قَالَ أَتَشَهَّدُ وَأَقُولُ اللَّهُمَّ إِنِّي أَسْأَلُكَ الْجَنَّةَ وَأَعُوذُ بِكَ مِنَ النَّارِ أَمَا إِنِّي لاَ أُحْسِنُ دَنْدَنَتَكَ وَلاَ دَنْدَنَةَ مُعَاذٍ . فَقَالَ النَّبِيُّ صلى الله عليه وسلم: حَوْلَهَا نُدَنْدِنُ
Abu-Salih reported on the authority of some Companions of the Prophet ﷺ that the Prophet (ﷺ) said to a person: "What do you say in prayer?"
He replied: "I first recite tashahhud, and then I say: O Allah, I ask Thee for Paradise, and I seek refuge in Thee from Hell-Fire, but I do not how to make it sound like your sound and the sound of Mu'adh [what you say or he says in prayer]."
The Prophet (ﷺ) said: "We too go around it [paradise and hellfire]."
— Sunan Abi Dawud » Book of Prayer (Kitab Al-Salat) » Hadith 792
Finally, there is a benefit that befalls us when we pray for others, especially while not in their presence. Angels affirm our supplication, and ask for the same to befall us:
حَدَّثَنَا إِسْحَاقُ بْنُ إِبْرَاهِيمَ، أَخْبَرَنَا النَّضْرُ بْنُ شُمَيْلٍ، حَدَّثَنَا مُوسَى بْنُ سَرْوَانَ، الْمُعَلِّمُ حَدَّثَنِي طَلْحَةُ بْنُ عُبَيْدِ اللَّهِ بْنِ كَرِيزٍ، قَالَ حَدَّثَتْنِي أُمُّ الدَّرْدَاءِ، قَالَتْ حَدَّثَنِي سَيِّدِي، أَنَّهُ سَمِعَ رَسُولَ اللَّهِ صلى الله عليه وسلم يَقُولُ: مَنْ دَعَا لأَخِيهِ بِظَهْرِ الْغَيْبِ قَالَ الْمَلَكُ الْمُوَكَّلُ بِهِ آمِينَ وَلَكَ بِمِثْلٍ
Umm Darda' reported: My husband reported that he heard Allah's Messenger (ﷺ) as saying: He who supplicates for his brother behind his back (in his absence), the Angel commissioned (for carrying supplication to his Lord) says: Amen, and it is for you also.
— Sahih Muslim » Remembrance of Allah, Supplication, Repentance and Seeking Forgiveness » Hadith 2732 b
