The ancient custom of learning ones Tehillim with Rashi’s commentary on Rosh Chodesh:[1]

It is an ancient custom on every Rosh Chodesh for one to study one verse from the chapter of Tehillim that corresponds to his age[2], together with the commentary of Rashi, [in a way that he completes its study on the Rosh Chodesh prior to his next birthday]. [On the first Rosh Chodesh after one’s birthday, one begins learning his new Kapitel of Tehillim with Rashi, and concludes it on the last Rosh Chodesh prior to his birthday.]

Other commentaries: If one chooses, he may study other commentaries as well, aside for the commentary of Rashi.

How many verses to learn each Rosh Chodesh-Kapitel has less than 12 verses: If the Psalm comprises fewer than twelve verses, or it is a leap year [in which even 12 verses do not suffice, and one requires a minimum of 13 verses in his Kappitle], then one repeats this study in subsequent months in such a way that the verses will match the number of months in that year. [For example, if it contains 10 verses, then, after studying the 10 verses in the first 10 Roshei Chodashim, one is to repeat the study of the verses on the last two Roshei Chodashim.[3]]

How many verses to learn each Rosh Chodesh-Kapitel has more than 12 verses: If the Psalm contains more than twelve verses, one studies two or more verses on each Rosh Chodesh. [For example, if it contains 20 verses, then, by the first 8 Roshei Chodashim one should learn two verses, and learn a single verse on the last four Roshei Chodashim.[4]]

The following is a link to study your chapter of Tehillim with Rashi in English: https://www.chabad.org/library/bible_cdo/aid/16222/showrashi/true

Two days of Rosh Chodesh:

How to calculate the two days of Rosh Chodesh: This matter is inconclusive. Perhaps when there are more than 12 verses, one should split up the verses in accordance to the number of two day Rosh Chodesh of that year. Alternatively, one should repeat the study of the same verse on both days of Rosh Chodesh and hence only split it to 12. [Vetzaruch Iyun, as from the letter above it is implied that we do not take the two days of Rosh Chodesh into the calculation at all, and that it counts as only one day.]

What should one learn on a two-day Rosh Chodesh? Seemingly, one is to learn a verse from his Tehillim with Rashi’s commentary on both days. Regarding if one should learn a different verse on each day, or simply repeat the verse learned the previous day, the implication from the above letter is that one repeats the same verse on the second day as was learned the first day, however, perhaps if there are more than 12 verses, then one should learn a new one each day. Vetzaruch Iyun, as stated above.

Does the above custom apply also on Rosh Hashanah?[5]

Yes, on the first day of Rosh Hashanah.

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[1] Sefer Haminhagim p. 69 [English]; Letter of Rebbe Rayatz printed Koveitz Michtavim in Tehillim Yosef Yitzchak, and in Igros Kodesh Rayatz 5:240; Luach Devar Beito; Rivivos Efraim 2:135-13

[2] For example, a thirty-year-old studies Psalm 31

[3] It is unclear as to what how many verses one is to repeat in such a case, and if he is simply to repeat the study of one verse each remaining month [i.e. verses 1 and 2], or if he is to repeat the study of all the verses split between the two remaining months [i.e. verses 1-5 the first remaining month and 5-10 the second remaining month], or if he is to repeat the first verse by the first three months and leave 9 verses for the remaining 9 months.

[4] It is unclear as to what how many verses one is to repeat in such a case, and if he is simply to repeat the study of one verse each remaining month [i.e. verses 1 and 2], or if he is to repeat the study of all the verses split between the two remaining months [i.e. verses 1-5 the first remaining month and 5-10 the second remaining month].

[5] So is implied from the letter of the Rebbe Rayatz who corresponds the saying of 12 verses to the 12 months, hence including Rosh Hashanah in the calculation. Now, although we don’t explicitly mention Rosh Chodesh on Rosh Hashanah in order to confuse the Satan, and due to other reasons [see Admur 591:6], nonetheless, since here Rosh Chodesh is not explicitly mentioned, and it is not part of any specific prayer, therefore there is no reason to avoid fulfilling the custom then.