The Meor Einayim teaches that אדר stands for דר 'א, the Alef, (the עולם של אלופו, Hashem) dwells. In Adar we can discover that Hashem dwells in this world and leads it with hashgachah pratis. The avodah of this month is to keep our eyes open and to recognize the דר 'א, that Hashem is here, with us, saving us, and helping us all the time.
The Ahavas Shalom zt'l says that אדר is roshei teivos for אשתכח דרעוין רעוא, "The desire of all desires prevails." The expression רעוא דרעוין means a time when Hashem's love and desire for us is expressed and felt profoundly. For example, shalosh seudos is considered to be the peak and holiest moments of Shabbos, and it is called דרעון רעוא. This love and desire to us is revealed in this month.
The custom of the tzaddikim of Ziditchov was that they wouldn’t say tachanun the entire month of Adar. The Yeshuos Yaakov zy'a (the Rav of Lemberg) met with one of the tzaddikim of Ziditchov (Rebbe Isaac of Ziditchov zy'a) and said, "Give me a source for this custom of not saying tachanun. And please, don't tell me a source from kabbalah or chassidus. I want a halachic source." Rebbe Isaac Ziditchover replied that it states (Esther 9:22), לשמחה מיגון להם נהפך אשר והחודש טוב ליום ומאבל, "The month that was transformed for them from grief to joy and from mourning to a yom tov..." So we see that the entire month became a yom tov, and we don’t say tachanun on a yom tov."
It is good to say Tehillim on Rosh Chodesh Adar. Reb Sariyah Dibletzky zt'l (Davar Nechmad) writes, "Reb Eliyahu Cohen and Reb Chaim Abulafia zt'l heard from Reb Chaim Vital's students that one... should recite the entire Tehillim on Rosh Chodesh Adar. This will fix his neshamah. There are other secrets, too. It is proper that those who fear Hashem should gather to make this tikkun."
The rule in spiritual matters is that if you can't do everything, do as much as possible. So, those who cannot say the entire Tehillim on Rosh Chodesh Adar should say a few chapters of Tehillim.