If I activate two shrines in quick succession, do I get the full effects of both for their respective durations, or does the last shrine buff supplant all previous shrine effects?
If the shrines are of different type (e.g. Fortune Shrine and Enlightenment Shrine) then yes, both effects can be active at the same time. They will both be listed as separate buffs above your spellbar.
However, if the shrines are of the same type, the second shrine effect will simply overwrite the first one and refresh the time left for the buff to the maximum.
One thing to note: In Act IV there are wells which give you frenzy and also heal you. They work the same way as normal shrines and don't stack.