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.