My question has to do with confusion over two different models of the
universe.  In one model, the universe is expanding at an accelerating
rate.  We can tell because the farther a galaxy is away from us, the
more red shifted it is.

The second model used to be favored until recently.  The universe
explodes in a big bang, galaxies fly away from each other, but gravity
pulls them back together over time until the expansion stops and the
universe ends ultimately in a "big crunch".  This could ALSO be
interpreted by observation because the farther we look into the
universe, the farther back in time we are looking.  Therefore, the
younger the universe was the faster galaxies were moving away from
each other.

The question is, what model is correct, and how do we know for sure?

If we measured the change in red shift over time in a given galaxy
this would confirm that the expansion is accelerating - not slowing
down.  Is it possible to measure the change in red shift over time in
a given galaxy or is it too small?

Those are not two different models.

The standard 'Big-Bang' model, based on General Relativity, and observed
red shifts of galaxies, allows three theoretical cases, hyperbolic, elliptic and
parabolic. 
As  iang-ga  explains in his comment.

 These have different 'average energy density' and one of them leads
to 'Big Crunch', others do not. Which case is realized, we are not sure.

General description is here: 
http://www.damtp.cam.ac.uk/user/gr/public/bb_home.html

Specific details are  here: 

 In most cases relativistic geometry M4 can be reduced to ordinary
Euclidean geometry and time (M4 --> E3+ E1) . If we do that with
current  mainstream
SEARCH TERM : Big Bang
model, we get the model you mention: Mostly empty, infinite space E3,
in which matter is flying apart, more and more slowly. It may stop and
start falling back (elliptic case), keep slowing down to zero,
(parabolic case) or, if it has enough kinetic energy to overcome the
gravity (hyperbolic case) it may keep expanding. We do not even know
which case we have - as sky surveys come too close to parabolic - to
zero energy case - to decide.
http://answers.google.com/answers/threadview?id=460093

In particular, this:
 " Therefore, the younger the universe was the faster galaxies
 were moving away from each other"
is not part of accepted models. Original, simple model was that expansion is
slowing down, about the same way (or rather similar way) as if you
shoot a satellite
into space. It will be slowing down, and depending on initial mass/energy, it may
return (elliptic case) or keep flying away (hyperbolic case).
With "inflation" and 'dark matter' hypotheses it became more complex.

You stated;
"We can tell because the farther a galaxy is away from us, the
more red shifted it is"

I always thought redshift determined the speed it was travelling away
from the measurement source, not the distance.  Anyway, as far as
proving which one is valid, it seems like a question that would be
difficult to answer without a time machine, or a teleporter to catch
the light from the 5 billion year old explosion still moving through
space.

The hubble has been rumored to have the ability to see 16 billion
light years distant.  Maybe it's possible to tweak the settings on a
telescope like that and catch the birth of our universe.

Just a random thought.

All I know on the subject I learned from Scientific American, http://www.sciam.com/

They have a search feature. Search arguments should include
'cosmology', 'strings' and 'multi-verse'. The observations now
indicate the universe not closed (i.e., it will expanded forever until
it becomes a cold collection of burned-out cinders).

The hot topic is string theory, which requires up to six additional
dimensions, and the notion of a 'multi-verse', that our universe (or,
sub-universe, actually) is one of what amounts to an infinity of other
sub-universes, where the physical laws can and probably are different
from those in our own.

The critical thing to know is the density of the universe or how much
mass/energy there is - this is known as Omega.

If you start with the big bang there are 3 possible futures  - 

In the first, gravity wins, the expansion of the universe is reversed
and everything comes back together - the Big Crunch. Omega is greater
than 1

In the second, the big bang wins. There's not enough mass/energy in
the universe, and hence not enough gravity to overcome the expansion
and the universe expands for ever.  Omega is less than 1

In the third it's a draw!  The expansion comes to a halt, but it takes
infinite time.  Omega = 1

So all we need to do is measure Omega :-)

Tied to the big bang is another theory (it's a model realy, but let's
not get too hung up on the detail) called Inflation, and this predicts
that Omega should be 1. It still needs some experimental verification
though, and you can do this by looking at the shape of the universe. 
A full explanation means getting into non-Euclidian geometry, but the
bottom line is that you can look at Cosmic Microwave Background
experiments like COBE and Boomerang and show that the universe has the
right shape for Omega to be 1.

If I'd written this 7 years ago I'd have stopped there, but in 1998
Saul Perlmutter measured the red-shifts of very distant supernovae and
showed that the expansion of the universe is accelerating.  What's
causing this acceleration isn't known, but it's been called dark
energy and understanding it is one of the big questions facing
cosmology.  How this affects the fate of the universe is open to
question too, though some theories have the rate of acceleration
increasing to the point that the universe is torn apart in a "Big
Rip".

You asked "...how do we know for sure". We don't!  That's what makes
it so much fun!

Ian G.

A tip of the hat for Ian G.'s excellent comment.

That may all be true, but when we are looking millions and billions of
light years out into space, we are looking at what was happening that
long ago, and have no idea of what has happened in the meantime.  The
stars and galaxies in "deep space" that were accelerating away from us
have all gone their way, disappeared into black holes, or whatever.
Could it be that the universe is already contracting?  Ian presents
the (an) argument against this happening, which I cannot argue
against, but we don't know, as he so cheerfully admits.  :-)
Myoarin

Of course, all this assumes that redshift is due expansion. Other
models explain redshift as the photons losing energy as they travel
along by bumping into electrons. This is supported by the fact that
measured values of the Hubble constant (64 km/s per Mpc) are exactly
equal to hr/m for the electron in each cubic metre of space. See
http://www.lyndonashmore.com
If this is the case then the universe is not expanding all!