acquire fat stacks of shiny, donate.
acquire fat stacks of meat and milk, donate.
use Extremely Hallowed Shrine Powder before large donations to triple their value.
Donate, donate, donate, donate, donate.
Observe that there is a cap on favor gained per giant per day: the cap is whatever value of favor you started at for the day (say, if you were 0/1000 for the day, your cap is 1000). It does go up by another 100 if you get an emblem (so if you get to 1000/1000, get the emblem, and are now 0/1100, you can donate another 100 worth).
These caps are exclusive of the XP cap, which is not "per giant" but for total donations.