But they are so far away and so far apart that to our eyes they appear to be fixed in space (at least, over our lifetimes). However, through some very tedious and careful techniques, the details of these motions has been worked out (at least for some of the nearest stars). And of course, the motions must be always measured relative to our own sun, which is also moving through space. The "apparent positions" of stars in the field of view of a telescope never changes (at least over the course of your lifetime). They all act as if they were rigidly "fixed" on the celestial sphere, and not moving relative to one another. However, if one of them is much closer to you, by shifting your point of view back and forth it appears to shift back and forth across the background.

I'm not too familiar with modern methods of astrometry, but when I was a student it was done photographically. Star positions on modern photographs were compared with carefully measured positions taken from old photographs. These measurements were made with high precision micrometers and microscopes, using dimensionally stable glass photographic plates taken many decades apart. But this only gave us the stars 'proper motion', its movement across our line of sight. A star's motion toward or away from us was determined in a completely different way, by Doppler measurements of its spectrum. Unscrambling all these different measurements and allowing for our own sun's orbital path around the galaxy was the problem.

Remember, the "motion" of an object is a meaningless term. Its always the relative motion between two objects that we measure--there is no preferred coordinate system in space. All motion is specified as relative to something else. There are techniques to measure motion in a coordinate system, but these reference frames must be defined by either a physical object (like the earth) or some relationship between physical objects (like the Vernal Equinox, where ecliptic crosses equator). And these are also in motion.

Measuring the distance to an object can be done only by triangulation, as is described in the article. But this method is limited to only the nearest stars due to the technical difficulty of measuring very tiny angles. However, we can make good estimates of greater distances if we apply our knowledge of the nature of these objects. For example, if we know stars of a certain spectral class are of a certain true brightness, then by measuring the apparent brightness of a star we can make a good estimate of its true brightness, and hence, its distance. But these are, at best, estimates, and the distance scales must be calibrated eventually on some actual measurement. Everything eventually relies on those stars actually close enough to show a measurable parallax.

When you look up a star's position in a catalog, or scale it off a chart, you come up with a pair of coordinates, The Declination and Right Ascension. DEC is the number of degrees N or S of the equator, RA is the number of hours east of the Vernal Equinox. But due to precession, the Vernal Equinox is in constant motion as the earth wobbles every few thousand years on its axis. Right now, catalog positions are referenced to where the Equinox was in the year 2000.0. But if you should need an accurate position for today (like to point your telescope) you need to do the math and
"precess" the catalog values to the present day.

And remember, a "position" isn't really a position. Its only a direction. You may know exactly where in the sky a star's direction is, but that doesn't necessarily mean you have any idea of how far away it is.