How big bang could produce later collisions The Milky Way and the Andromeda galaxies are supposed to "collide" in 4 billion years (collide in the sense of overlapping space, but nothing is really supposed to contact anything else).
Assuming:


*

*All matter in the known universe was ejected from a single point an
exceedingly long time ago

*Objects in motion tend to stay in motion in a straight path (See
Newton's Laws)


How could these macro objects collide?
If both galaxies were traveling on the same path (one slower than the other), what would cause it to suddenly start speeding up?
If both galaxies left the original point in slightly different directions, how could they change their paths to intersect with each other? I'm guessing there aren't any giant walls for galaxies to bounce off of?
Edit: I am not really asking if the universe started at a point. I'm really asking about how a giant expansion could lead to paths where large exotic objects could collide into each other. Doesn't seem duplicative to me.
 A: First of all, the big bang did not happen at a single point. There is a possibility that the universe is infinite: in that case it was infinite already at the moment of big bang.
Now how can there be objects approaching each other, when everything was expanding from the beginning? The thing is that it's been more than 13 billion years since the start of the expansion and the current value of the Hubble constant is 67.8 km/s/Mpc. It means that the expansion rate is very small for objects at distances below Mpc (67.8 km/s is not a lot) and the gravitational interaction dominates at the scales between Mpc and the meaningful range of other known interactions. That is why the Solar system is not expanding and apples are falling towards the Earth.
The motion of galaxies inside the Local Group is also ruled mainly by gravity. The galaxies are orbiting the common center of gravity and the expansion of universe is not affecting that too much. Because the motions of galaxies in a cluster can be quite complex, it just happens that sometimes some galaxies can approach each other inside one galaxy cluster.
This would of course not be possible at the very early stage of the universe evolution, when there were no galaxies and the expansion was dominating even at small distances (see for example the discussion here). But this is no longer the case. Local deviations from the expansion are quite common today.
A: This is really an extended comment to mpv's answer.
Observations of the microwave background tell us that 400,000 years after the Big Bang the universe was extremely smooth. The density differences were of order 0.001%. To a very good approximation everything was moving smoothly away from everything else and, as you say, collisions wouldn't be happening.
However a smooth distribution of matter is unstable. If some region has a density even slightly greater than average that region will start to collapse and become denser under its own gravity. Likewise regions that are lower than average density will become increasingly less dense. This process is how matter clumped to form the galaxies and stars that we see around us today.
So if we consider some giant (galaxy supercluster mass) expanding dust cloud in the early universe it starts out expanding with every bit of it moving away from every other bit:

But once the cloud has started collapsing this means the edges of the cloud must be moving towards each other. The velocity of the matter at the edges of the cloud is now different from the average velocity due to the expansion of the universe. Technically we say the matter at the edges of the cloud has developed a peculiar velocity due to the gravitational forces acting in the cloud.
It would be tempting to immediately conclude that the Milky Way and Andromeda are approaching each other because they were originally on different side of a collapsing dust cloud, but this is almost certainly untrue. The galaxy distributions we see today are the result of a prolonged evolutionary process. The original dust clouds fragmented and fragmented again to form galaxy clusters, groups and finally individual galaxies, and all these objects interacted gravitationally in complicated ways. However the basis principle remains that the peculiar velocities we observe today are the result of gravitational forces pulling matter away from the original smooth expansion.
A: Your two assumptions are within the classical Newtonian gravitation/mechanics.

All matter in the known universe was ejected from a single point an exceedingly long time ago

The observations leading to a simplistic  Big Bang model still talks of a General Relativity solution. Such a hypothetical singularity is not a point in three dimensional space to follow the Newtonian mechanics of an explosion. All present space points were at that original singularity.

Objects in motion tend to stay in motion in a straight path (See Newton's Laws)

Yes, in Newtonian physics.BUT the cosmological model now still called Big Bang is a General Relativity model, not a Newtonian model.
Here is the cosmological story built up by the observations up to the present.

The inflation period, right at the beginning of the story, homogenizes the available energy with very small  inhomogeneities , seen  in the Cosmic Microwave Background (CMB) radiation, which gives a snapshot of the universe   at about 380.000 years after the beginning.

These inhomogeneities are the sperms which will develop into the granular present universe, which is composed of matter concentrated into the galaxies  and clusters of galaxies we observe now. At their very creation there was an energy and momentum associated  with each of the blobs in the image , i.e. there could be possible motion for merging or against, initiated by the quantum fluctuations of the inflation period. 
Since then, as other answers state, the gravitational attraction from concentrations of matter would also play a large role. At the present time, the modern universe in the figure,  Newtonian physics is a good approximation of the kinematics of the firmament , and will answer any questions.

How could these macro objects collide?

They might have started on a collision course from the original inflaton stage. It may be the gravitational attraction that started the trajectories.

If both galaxies were traveling on the same path (one slower than the other), what would cause it to suddenly start speeding up?

To start with, they were not galaxies but concentration of primordial matter/energy responding to gravitational forces. 
Speeding up and slowing down would come from the many body gravitational interactions in the neighborhood of each galaxy. One would need the specific boundary conditions to fit the observation with a newtonian trajectory.

If both galaxies left the original point in slightly different directions, how could they change their paths to intersect with each other? 

Again , gravitational attraction can overcome and change direction of velocities. After all that is what is holding the moon around the earth and not on it :), or flying away.
