How do I go about starting to show that, if G acts on X and x is in X, with y=g(x) for some g in G, then the stabiliser of y is the conjugate of the stabiliser of x?
you know that y = g(x). it follows that g^-1(y) = x.
now suppose that a is in Stab(x).
then gag^-1(y) = g(a(g^-1(y))) = g(a(x)) = g(x) = y.
thus gag^-1 is in Stab(y).
similarly, if b is in Stab(y), then g^-1bg is in Stab(x), so
b = g(g^-1bg)g^-1 is in g(Stab(x))g^-1.
from this we can conclude that g(Stab(x))g^-1 = Stab(y).
(we only need the first part if G is finite, since then
g(Stab(x))g^-1 contained in Stab(y) shows they are equal).