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What halachos of mourning, if any, apply to a grandchild upon his or her grandparent's death? Do they tear keri'ah upon hearing of the grandparent's death? Do they sit shivah?

Upon hearing bad news, one says the blessing ברוך אתה יקוק אלוקינו מלך העולם דיין האמת"‏" (OC 222:2).

Regarding mourning, the Shulchan Arukh rules (YD 374.6):

כל מי שמתאבל עליו מתאבל עמו אם מת לו מת ודוקא בעודו בפניו אבל שלא בפניו אין צריך לנהוג אבלות...: הגה וי"א דהאידנא נוהגין להקל באבילות זה של המתאבלים עמו דאין זה אלא משום כבוד המתאבלים ועכשיו נהגו כולם למחול וכן נוהגין האידנא שלא להתאבל כלל עם המתאבלים וכל המחמיר בזה אינו אלא מן המתמיהים (הגהות מיימוני פ"ב) ומ"מ נהגו שכל קרובי המת הפסולים לו לעדות מראים קצת אבילות בעצמן כל שבוע הראשונה דהיינו עד אחר שבת הראשון שאינם רוחצים ואין משנים קצת בגדיהם כמו בשאר שבת ויש מקומות שנהגו עוד להחמיר בענינים אחרים והעיקר כמו שכתבתי ... מי שרוצה להחמיר על עצמו להתאבל על מי שאינו צריך או ללבוש שחורים על קרובו אין מוחין בידו

Whomever he mourns for he [also] mourns with if he suffers a breavement. [This applies] only whilst he is in his presence, but when not in his presence, he is not bound to observe mourning rites... Gloss: Some say that nowadays we adopt a lenient view regarding this [type] of mourning with respect to those who mourn with him, for this is only out of respect to those who observe mourning rites, but now it is the common practice to forego [this honour], and thus is the accepted practice nowadays, [viz.,] not to mourn at all with those who observe mourning rites. He who adopts a stringent view in this [matter] is one of those who make people wonder [at their strange conduct]. Nevertheless, it is the accepted practice that all the near-of-kin of the dead person who are disqualified to offer testimony in his behalf, exhibit some [signs] of mourning [by placing] themselves [under this stringency] the first entire week, i.e., until after the first Sabbath [in] that they do not wash [themselves] nor change some of their clothes as on other Sabbaths. There are, however, localities where the accepted usage is to adopt additional stringencies in other matters [pertaining to this type of mourning], but the fundamental principle is as I have written [above]... One who desires to place himself under a stringency, [viz.,] to observe mourning rites for one for whom he is not dutibound [to mourn], or to put on dark clothes [in mourning] for his near-of-kin, is not reproved [for doing so].

Regarding tearing clothing, the Shulchan Arukh rules (YD 340:4):

כשם שקורע על קרובו שמתאבל עליו כך קורע בפני קרובו על מת שמת לקרובו כיצד הרי שמת בן בנו או אחי בנו או אם בנו חייב לקרוע בפני בנו וכן קורע על חמיו וחמותו והאשה קורעת על חמיה וחמותה. (ואין נוהגין עכשיו כן וכמו שיתבאר לקמן סי' שע"ד לענין אבלות):‏

Just as he rends for his near-of-kin, for whom he observes mourning rites, so does he rend in the presence of his near-of-kin, when the latter sustains a bereavement. In what respect? If his son's son dies, or his son's brother, or the mother of his son, he is required to rend in the presence of his son. So too, does he rend for his father-in-law and his mother-in-law, and a woman rends for her father-in-law and her mother-in-law. (Nowadays we do not follow this practice as will be explained infra §374 in the subject of mourning.)

In short, if the mourners insist the grandchild would have to act like a mourner in pretty much all respects while in front of them. Commonly though the mourners do not insist. Use of common sense to fit in and not offend the mourners is key.

(I'll say that even stronger: the grandchild is not in the thick of it like his parent is and is in a unique position to offer much needed support, so helping his parent in whatever way works best in their situation is a enormous act of kindness and is to be actively encouraged. Forget about the technical laws and do whatever is best for the real mourners which might indeed mean looking distraught and not shaving or bathing, but might not. Focus on the goal here not on the means.)

You can find some discussion in Rishonim trying to obligate grandchildren in particular in more mourning due to the obligation to honor a grandparent in Or Zarua 2:428.