Not to stare excessively at a rainbow:
It is forbidden for one to stare excessively at a rainbow. [However, it is permitted to look at it without an intense stare or prolonged contemplation. The rainbow corresponds to the Shechina, and hence one who looks at it is considered to be looking at the Shechina. The Talmud states that one who looks at it excessively is considered to not care for the honor of His Maker, and it is thus befitting for him to not have come into the world. Likewise, one who does so causes his eyesight to weaken. The Zohar states that there are Kelipos which surround the rainbow and try to darken its light, thus explaining another reason for why it should not be looked it.]
Not to tell others of the rainbow: One who sees a rainbow is not to inform others of it, as doing so is considered slander. [If one was informed of its presence, he has no obligation to go see it in order to recite the blessing.]
Q&A on the staring
May one take a picture of a rainbow?
Seemingly, one is to avoid taking a picture of a rainbow, as it is equal to staring at it for the sake of study, which is forbidden to be done. Likewise, publicizing the picture is no different than the warning brought above against telling others. Vetzaruch Iyun.
Is one to avoid staring at the reflection of a rainbow seen within water?
Some write that one is to avoid doing so.
The message of the rainbow:
The purpose of the rainbow is to arouse Divine mercy upon the world and grant us protection. The Zohar states that when the rainbow is seen, it is a sign that the Reshaim have reached the point of needing destruction, and our forefathers then Daven on their behalf to save them.
How can the rainbow serve as a message from G-d if it is embedded within nature?
Several answers have been offered by the Mefarshim:
1. There are two types of rainbows, one being the natural rainbow and the second being the rainbow which Hashem placed as a sign for the covenant.
2. Hashem could arrange for it to rain only at night, hence causing the rainbow to never be seen.
3. Hashem to begin with planted the rainbow within nature because he was aware of the covenant that he would make.
 Michaber 229:1; Chagiga 16a as explains Rosh and Tur
 Beis Yosef 229 in name of Rosh “The prohibition is only to stare as one who stares focuses on all the details in a much greater fashion than one who simply looks”; Biur Hagr”a “One must see it in order to say the blessing and Lashon Histaklus means with great contemplation”; Machatzis Hashekel 229:1; Olas Tamid 229:1; Kaf Hachaim 229:5
 Zohar Beshalach 66b
 Rebbe Aba in Chagigah ibid; Machatzis Hashekel ibid
 M”A 229:2; Chagigah 16a [see Machatzis Hashekel ibid and P”M 229 A”A 2 who replace the word Shlah in M”A ibid]; Olas Tamid 229:1; M”B 229:5; Kaf Hachaim 229:5
 Tikkunei Zohar Tikkun 18 p. 36b
 Chayeh Adam 63; Pischeiy Teshuvah; M”B 229:1; Kaf Hachaim 229:1
 Makor Chaim Kitzur Halachos 229; Piskeiy Teshuvos 229:3
 Yalkut Hagershoni 229:2; Ruach Chaim 229; Piskeiy Teshuvos 229:2
 See Piskeiy Teshuvos 229:2 footnote 9
 Rav Yonason Eibashitz in Yearos Devash 1 Derush 12
 Shlah Parshas Noach in name of Rama
 Shlah Parshas Noach