There are two elements to why the universe appears to be so orderly: the physical laws of that govern the universe are the same everywhere, and astronomical objects are very, very, very far from each other.
Consider two objects, one much larger than the other, and both very far from anything else. Because of gravity (which works the same everywhere), the smaller will either move in an ellipse around the larger, or fly by following a hyperbolic trajectory, and disappear into the distance. The exact starting position and velocity will affect only the particulars of the ellipse or hyperbola, but any set of starting conditions will result on one of those two.
Now consider the solar system. If you take something (much smaller than the sun) and stick it somewhere at random in the solar system and give it a random velocity, chances are very good that it will follow a hyperbolic trajectory or an elliptical orbit, because everything else is so widely spaced that it is unlikely to come close enough to anything else for it to matter much: the situation is very probably almost like the two-object scenario above, and the resultant path for the object is very probably almost what it would have been in that scenario.
Of course, the "very probably" and "almost" here are important. There are plenty of exceptions where objects pass near objects other than the sun, and you need to take the gravity of Jupiter and other planets into account to calculate paths to high precision.
If you look at objects where no atmosphere is present to wear away the evidence (eg the moon, Mars), you see plenty of evidence that there have been plenty of collisions, and of course these collisions are still ongoing (eg Schumacher-Levy 9).
In systems that aren't so simple, such as star clusters or our galaxy, the situation is more complex. The main reason why galaxies and star clusters look so orderly is that the distances between the stars is so vast that, even when the stars are moving very fast, the change we see in their overall pattern over the course of a human lifetime is very slight.
Even over longer timescales, though, the density is small enough that there are few collisions or even close interactions (see Binney & Tremaine's book, Galactic Dynamics, pp. 187-190). Instead, stars follow a potential due to the collective gravity of all the matter in the galaxy. In a roughly spherical system, this might result in roughly elliptical orbits, but might also result in rosette like (unclosed) trajectories (see Binney & Tremaine pp103-110). In a system were most interactions are gravitational, asymmetries in the distribution of distant stars are just as important as nearby stars. (Although the gravity of a nearby mass drops as the distance squared, the amount of matter at a given distance increases as the distance squared.)
The specific orbits of stars within these potentials are not particularly ordered. You can see this if you compare the the smoothness in the distribution of young stars with those of old ones. Young stars tend to form in clumps ("star forming regions," places where there is a gas cloud under the right conditins for star formation, eg LH 95, IC 5146), so galaxies with lots of young stars tend to have visible structure (sometimes messy like I Zw 18 and NGC 4214, sometimes not, as in NGC 5248) the details of which depend on the dynamics of the gas in the galaxy. Over time, though, only the densest clumps remain together (because of their mutual gravity); otherwise, the variety of trajectories taken by different stars from the same star forming region will spread it out over time. Galaxies with mostly very old stars, like M 87, therefore, tend to be mostly very smooth, with a population of very dense clumps (globular clusters) which show up in our images as point sources because they are so far away. (Globular clusters in our own galaxy can be resolved, spectacularly, into individual stars; see, for example, M 13 and M 3.)
Interestingly, the underlying randomness (disorder) in the trajectories of stars leads to interesting instances of apparent order. Just as the random motions of molecules in a gas allow us to use statistical laws to make precise descriptions of the behaviour of gas, the random trajectories of stars in globular clusters results in a surprising uniformity in their appearance. See this paper by Madsen. (Globular clusters are old enough and compact enough that interactions between individual stars can actually be important; see Binney & Tremaine p190).
On even larger scales, dramatic interactions between galaxies are quite common. NGC 3227 is a nice example; more can be seen here. In the case of smaller galaxies merging with the Milky Way, we can see the different trajectories of individual stars from the smaller galaxy spreading them out more smoothly over our galaxy. Several of these seem to be going on at once in the "Field of streams".
