I am working on the model which can visualise how the night sky change over time (from observer's point of view). I was thinking how axial precession affect positions of stars and planets on the night sky, but after hour of straight thinking I can't put it together. I use as reference point of view the north pole. Lets say that axial precession is circular (following circle) and we are looking at the sky at the same part of the year. If I understand it correctly, the stars/planets will move in two ways - in circle and down from the original position. So in the quarter of axial precession cycle, stars which were originally on the norhtern sky, are now on the eastern sky (I read that axial precession proceed in the opposite way as rotation of the planet) and sky should appear as if planet doesn't have axial tilt (which should make planet without seasons) - in different words, angle between star's original position and current position is the same as axial tilt. In the half of axial precession cycle, stars which were originally on the norhtern sky, are now on the southern sky, angle between their original position and current position is double of axial tilt. That's my understanding of axial precession. But apparently (from what I read) axial precession just shifts seasons (and doesn't stop them in the quarter of cycle) and I also run some simulations in Stellarium and stars with closer declination to the equator don't change their declination as much as stars near the north pole. I really don't know what I am missing.
How axial precession affects the night sky
For layspeople, there are no discernible effects whatsoever. Even if they could magically live for millennia.
For the kind of nerd who is properly represented in media such as the tv show The Big Bang Theory, the main observable effect on the night sky is that the polar star changes every few thousand years. It was Alpha Draconis back at some point when ancient Egypt was not ancient. It is Polaris now, and will be Gamma Cephei when Valve releases Half-Life 3. All stars in the Northern hemisphere seem to circle around whatever star is the current polar one. There is no true southern polar star, so observers there wouldn't notice much regarding this.
The secondary effect is that the dates for equinoxes and soltices changes a little bit over time. As in approximately 13,000 years from now (both past and future), summer in the Northern hemisphere would be around December and the Northern winter would be in the middle of the year. The effect on the night sky are the dates when night are longer or shorter.
Also because of the above, the position of the Sun relative to the stellar background has a similar cycle. 13,000 years from now, you'll have to be born between May and June to be a floricking Sagittarius.
Over such vast spans of time, all stars other than those eligible to become polar change positions more due to their own trajectories around the Milky Way than because of Earth's movements.
What axial precession does is move the celestial north pole on a circle with a radius equal to the axial tilt. For Earth, a complete precessional cycle is completed in about 26,000 years. The entire starry sky appears to revolve 360° eastwards $\rightarrow$along the ecliptic$\leftarrow$ during a precession cycle. Year after year, the vernal equinox, aka the first point of Aries, appears to move westwards by about 50″ per year, or by about 1′ every 71½ years.
Saying that the vernal equinox appears to move westwards with respect to the stars is the same thing as saying the stars appear to move eastwards with respect to the vernal equinox.
For somebody who is at least a little bit familiar with the night sky, this has two very noticeable effects:
About 13,000 years from now, stars which right now are visible in summer (because they culminate at night) and not in winter (because they would be in daylight) will be visible in winter and not in summer; and vice-versa.
For people who live at mid-latitudes, the best example may be the striking constellation of Orion. Everybody who is even remotely familiar with the night sky can recognize it. Right now, for most Europeans, or Asians, or North Americans, Orion is a winter constellation, not visible in summer; but 13,000 years from now it will be a summer constellation, not visible in winter. (Of course, for Australians Orion is a summer constellation, and 13,000 years from now it will be a winter constellation...)
Of course, the effect is gradual, so that about 7,500 years from now Orion will be visible in spring in the northen hemisphere, roughly from mid-winter to mid-summer, but not in autum.
The twelve constellations of the Zodiac are, by definition, on the ecliptic. About half of them are visible in summer, and the other half are visible in winter. As the precession cycle progresses, about every 2,200 years one zodiacal constellation moves from the summer-visible group to the winter-visible group, and one moves from the winter-visible group to the summer-visible group.
This is the dawning of the Age of Aquarius... When the vernal equinox is in Aquarius this means that the constellation is visible in northern hemisphere autumn and southern hemisphere spring, and invisible, because it would be in daylight, in northern hemisphere spring and southern hemisphere autumn.
The position of the north and south poles in the sky changes quite fast on historical scales.
For example, about 2,300 years ago, the Greek explorer Pytheas of Massalia complained that there were no stars around the north pole. (In his days, the closest star from the north pole was β Ursae Minoris, sometimes called Kochab, which is a bright star, but was almost 10° away from the pole.) The north pole came within 5° of our current North Star, α Ursae Minoris, only about 1,000 years ago, and 1,000 years from now it will again be more than 5° away from it. (The closest approach will be around the year 2,100, when Polaris will be less than half a degree away from the north pole.)
For a quick reference, the full Moon is about half a degree across.
I have not said anything about planets. The planets will move relative to the stars as they always do, precession does not have any effect on this. The movement of planets is not synchronized with our years, so that the position of a planet is relative to the stars will not return to the same point after one year. (Of course, except the Sun, which returns to the same point with respect to the stars after one sidereal year by definition.)
About those equatorial coordinates:
The equatorial coordinate system, right ascension and declination, is equatorial. Axial precession rotates the starry sky around the axis of the ecliptic. As a consequence, the change in equatorial coordinates during a precession cycle is neither simple, not the same for all stars.
The ecliptic coordinate system comes to the rescue. Basically, ignoring the variations in the orientation of the ecliptic and the proper motions of the stars, the ecliptic longitude of a star goes through all the values from 0° to 360° during a precession cycle. For stars which are not very close to the poles of the ecliptic this comes to a regular increase in ecliptic longitute of about 1°23′48″ per century. Again ignoring the variations in the orientation of the ecliptic and the proper motions of the stars, the ecliptic latitude of a fixed star remains unchanged during a precession cycle.
So that one can always convert the equatorial coordinates of a star for a given epoch into ecliptic coordinates, apply the regular progress of ecliptic longitude, and convert back into equatorial coordinates for the target epoch.
Beware that the obliquity of the ecliptic, which is needed when converting between equatorial and ecliptic coordinates will change during a precession cycle. How to account for this depends on the time scale; on short time scales, a few centuries, we have very precise formulas, whereas on longer time scales this is governed by one of the famous Milankovitch cycles, varying between about 22.1 and 24.5 degrees in a cycle about 41,000 years long.