To summarize a long answer, space is very, very, very transparent. So some of the brightest appearing objects at only one point will be very, very, very far away and measuring the angles between them will give your position on a large scale. Measuring the angles between closer objects will narrow your position down, and so on.
When planning an interstellar voyage, people will aim their star ship at where the destination star will be when they get there, not at where it is now.
There are only few types of events that can destroy a star, and they can usually be predicted thousands, millions, or billions of years in the future, so it will be rare for space travelers to arrive and find their destination no longer exists.
And here are the details of my long answer.
When people shoot at a moving target at a shooting gallery, when hunting, or in war, they "lead the target". They estimate how fast the target is moving sideways and aim a little ahead of the target when they pull the trigger.
Many modern weapons systems have computers to calculate the trajectory of the target and adjust the trajectory of the missile to make sure that it will hit the target. And they do that more or less instantly.
The trajectory of the target star in an interstellar mission might be calculated over and over again over a period of decades to prefect the mission calculations before the mission ever starts.
Astronomers have been measuring the distances to some stars with ever increasing accuracy for over 180 years starting in the late 1830s. The method involves measuring the position of the star with great accuracy a number of times spaced 6 months apart, when the Earth is on opposite ends of its orbit. That makes the apparent position of the star wobble back and forth very slightly, which is called its parallax, and the amount of the parallax gives the distance to the star.
All stars are so distant that even the closest stars have parallaxes less than one arc second - and an arc second is only 0.0000007 of a full circle. Astronomers have been measuring angles of less than an arc second for more than 180 years.
From 1989 to 1993 the Hipparcos satellite of the European Space Agency measured the positions of over 120,000 stars with an average precision of about 0.001 arc second.
The Gaia satellite, also by the European Space Agency, has been measuring the positions of millions of stars and other objects since 2013. The goal is to provide a 3D map of about 1,000,000,000 stars, about 1 percent of all the stars in our galaxy. The accuracy of angle measurement is to be about 20 micro arc seconds.
When travel within our solar system is more advanced, satellites at least as advanced as Gaia will be placed in the leading and trailing Trojan points of the orbits of the four giant planets in our solar system. At any given moment the two satellites in Jupiter's orbit will be separated by about 5.2 times, the two satellites in Saturn's orbit will be separated by about 9.54 times, the two satellites in Uranus's orbit will be separated by about 19.22 times, and the two satellites in Neptune's orbit will be separated by about 30.06 times the total separation between points in Earth's orbit 6 months apart.
The precision of measurements should increase in the same ratio as the length of the baseline increases.
And if a faster than light drive is invented a bunch of manned or automated astrometric observatories will be placed in positions 1 parsec from the Sun in all directions. Since a parsec is 206,265 astronomical units (an astronomical unit is the distance from Earth to the Sun), two observatories on opposite sides from the solar system will have a baseline 206,265 times as long as observatories in Earth orbit or on Earth, and their measurements will thus be 206,265 times as precise.
How will the movements of the stars be discovered? The same ways they have been discovered for a century already.
The movement of a star, relative the the solar system, has two components.
One is the radial velocity toward or away from the solar system. The spectrum of the star will show a doppler shift that shows how fast it is moving toward or away from the solar system. If one knows how far away the star was when the light that reaches us was emitted, and thus how many years the light has traveled, and if the doppler shift in the spectrum tells how fast the star is moving toward or away from the solar system, it is easy to calculate how far away the star is now, or how far away it will be when your starship reaches it.
The other component of a star's motion is the sideways motion of the star relative to the solar system, the proper motion. That is detected by measuring the direction to the star several times over years and noticing any tiny change in the direction. Because the closest stars are likely to have the largest apparent proper motion, astronomers often selected stars with high proper motion for the first measurements of stellar parallax back in the 1830s and 1840s, so proper motion has been measured with increasing accuracy for at least 180 years.
And one of the main missions of the Hipparchos and Gaia astrometric satellites has been to measure the proper motion of many stars much more accurately than before.
So if a star is exactly 100 light years from Earth, it would be exactly 100 times 9,460,730,472,580.8 kilometers, or 946,073,047,258,080 kilometers from Earth.
If a starship can travel 100 times as fast as light for the entire journey, it will take it exactly one year of 365 days (light years are the distance light travels in 365 days) to reach the target star, at a speed of 0.2739726 light years per day, or 99.9999 light days per day - make it an even 100 light days per day, or 2,400 light hours per day, or 144,000 light minutes per day.
If the target star has a fairly reasonable sideways speed or proper motion of about 100 to 500 kilometers per second, it should travel 6,000 to 30,000 kilometers in a minute of 30 seconds, 360,000 to 1,800,000 kilometers in an hour of 3,600 seconds, 8,640,000 to 43,200,000 kilometers in a day of 24 hours or 86,400 seconds, and 3,153,600,000 to 15,768,000,000 kilometers in a year of 365 days or 8,760 hours, or 31,536,000 seconds.
Since a light minute equals 17,987,547 kilometers and a light hour equals 1,079,252,820 kilometers, a distance of 3,153,600,000 to 15,768,000,000 kilometers would equal 2.9220 to 14.4759 light hours distance, or 0.0012175 to 0.0060316 days travel time for the starship, or 0.02922 to 0.1447 hours travel time for the starship, or 1.7532 to 8.685 minutes travel time for the starship.
And that is only if the starship aims at the direction where the star is when it leaves, instead of aiming at the direction where the star will be a year later.
And what if a starship travels at only 1 percent of the speed of light to reach the star 100 light years distant? It will take the starship 10,000 years to reach the destination star, and in that time the proper motion of the star will move it 10,000 times as far to the side as in the earlier example.
Thus the target star will move about 31,536,000,000,000 to 157,680,000,000,000 kilometers, or 3.3333578 to 16.666789 light years, in 10,000 years, which will take the starship about 333.33578 to 1,666.6789 years to travel at one percent of the speed of light.
Thus the importance of calculating the future position of the star and aiming for that future position is proportional to the length of time that the trip will take.
So star ships will tend to aim for a future position of the star instead of its exact present position. The navigators will also be able to observe the apparent position of the star during the voyage, and if they notice any minor errors in the course calculations the ship can adjusts its course during the voyage.
If faster than light space ships "jump" from one point to another in space, without travelling the distance between them, then it seems simple to construct a formula to calculate the time a voyage from one point to another will take.
IMHO the formula should be: (X)Y + (X-1)Z, when X is the number of jumps made in the voyage, Y is the average time that the jumps may take, and Z is the average length of time it takes the ship to recharge its batteries, or recalculate, or for the crew to recover from the stress, or whatever, between each jump. Of course Y and Z can be zero, and X could be anything from one to infinity.
Within a glaxy, the stars that have the largest relative velocity to each other are likely to be on the opposite sides of the galactic center, since they will be travelling in opposite directions as they orbit the center of the galaxy. The Sun has an orbital speed of about 225 kilometers per second, so a star on the opposite side of the galaxy at the same distance from the center should have a velocity difference of about 450 kilometers per second relative to the Sun. Stars closer in could have orbital speeds of 1,000 kilometers per second, so two such stars on opposite sides of the galaxy should have a speed difference of about 2,000 kilometers per second, and so on.
The gravity between the Milky Way Galaxy and the Andromeda Galaxy is pulling them together at a speed of about 110 kilometers per second, and they are expected to collide in about 4,000,000,000 years.
Only about 100 nearby galaxies are approaching our galaxy. The vast majority of galaxies are moving farther apart due to the expansion of the universe. The farther away they are, the greater the velocity difference. Hundreds of kilometers per second, thousands of kilometers per second, tens of thousands of kilometers per second, and so on.
The farther away a distant galaxy is, the longer the light from it took to reach Earth, and the farther away it is now than when the light was emitted.
The oldest electromagnetic radiation detected is about 13,799,000,000 years old and thus was emitted about 13,799,000,000 light years from Earth. And during the 13,799,000,000 years it took that light to reach Earth, the places where it was emitted have moved much farther away from Earth. It is believed the source of that radiation is now about 46,500,000,000 light years from Earth.
So the distance between Earth and the source of that radiation has increased by about 32,701,000,000 light years in the last 13,799,000,000 years. So you can calculate that the distant sources of the oldest known radiation have been moving away from Earth at an average speed of 2.3698 times the speed of light, which is impossible. Actually the distant galaxies are not moving away from each other and Earth, the space between them is increasing in size, so the speed of light limit doesn't apply.
Anyway, that shows that very careful calculations would have to be made for a very long space voyage of billions of light years.
A given star might not even exist anymore because it has gone nova, but we simply can't know that before the light from the explosion reaches us.
Actually, stars do not just go nova at any random moment. Novae have causes, and astrophysicists can study a star and determine if it is ever going to go nova, and if so, approximately when. Rigel, or Beta Orionis, for example, is predicted to become a type II supernova in about ten million years.