Let's say I put the plane into a 30 degree bank with ailerons yet maintain
neutral rudder. To simplify things let's say I do this instantaneously.
Right at that moment, the nose is pointing in the original heading. But
what happens as this fairly large (earth referenced) horizontal lift force
acts on the plane; clearly it moves it sideways. But what happens to the
heading, that is messing me up.


The heading does not change. The force is pushing the aircraft sideways.

Well, ok, the vertical tail will encounter some (sideways) resistance, causing
the plane to weathervane a bit into the wind and changing the heading too, but
that effect is small for small banks.


If the force continues, either it results
in an acceleration which causes increasing
velocity or it reaches a terminal
velocity where the force is balanced by drag.


The latter, ultimately. And yes, the plane is then "climbing" sideways. Part
of the reason for the heading change you would fight is the tail, which
produces down lift, keeping the nose up against gravity. When banked, this
results in a turn. But then less is available to keep the nose up, which is
why you apply back pressure on the elevators.

So all these things are interdependent, and become more so as the amount of
bank is increased.

Taken to the extreme (an aerobatics pilot might chime in here), if the wings
are vertical (a knife edge) the only thing keeping the airplane up is the
(once) vertical stabilizer and the fuselage. The (once) horizontal tail will
want to keep the nose "up", which is sideways, and the plane will want to turn
(do a loop horizontally). Ignoring other effects, of course.

Jose

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