Scientists calculate how long it will take us to explore the galaxy

Spacecraft Pioneer 10 and 11 and Voyager 1 and 2, which were launched in the 1970s, are leaving the solar system.

What will happen to them? Will they reach other star systems? If so, how long will it take them? These figures may give us an idea of how long it could take humanity to get out of the reach of the solar system.

A couple of physicists, from the Max Planck Institute for Astronomy and the Jet Propulsion Laboratory at CIT, have discovered a way to estimate how long it would take a spacecraft like this to find its way to other star systems.

Using the Gaia Observatory's astrometric and radial velocity data, researchers integrated the trajectories of 7.4 million stars, and the spacecraft, through galactic potential, to identify those stars that the spacecraft will pass closest to. The most up-to-date map of Gaia has locations of about 7.2 million stars. They combined that data with the projected paths of the Pioneer 10, Pioneer 11, Voyager 1 and Voyager 2 space probes, launched between 1972 and 1977.

The conclusion was a bit discouraging

It could be tens of thousands of years before one of these probes approaches another star system and several million years before there is any direct contact."The time scale for a spacecraft collision with a star is of the order of 10^20 [one hundred quintillions] years, so the spacecraft has a long future ahead of it," experts write.

Physicists discovered that the four spacecrafts will approach approximately 60 stars for the next million years, and will meet within two parsecs with approximately 10 of them. They also discovered that Pioneer 10 will probably be the first to pass through a star system, one called HIP 117795. 

Their calculations show that the spacecraft will pass within 0.231 parsecs of the star in approximately 90,000 years. A reasonable distance (in cosmic terms).
To put it in perspective, Pluto's orbit takes it over 7 billion kilometres. So our probes will be tens of thousands of times that distance from alien stars.

In general, before they collide or are captured by a star system, spacecraft could be travelling about 1020 years. Gaia was launched by the European Space Agency in 2013 and parked at a point just outside of Earth's orbit around the sun. Since then, it has been collecting information about a billion stars, including their paths through space.
Future space missions do not have to follow the path of these four probes, of course: our closest star system, Alpha Centauri, is only 4.37 light-years or 1.34 parsecs away. At its current speed, Voyager 1 could get there in less than 80,000 years. Until scientists figure out how to develop a warp engine like Star Trek to move faster than the speed of light, we have a long wait.

Reference: Coryn A. L. Bailer-Jones et al. Future Stellar Flybys of the Voyager and Pioneer Spacecraft, Research Notes of the AAS (2019). DOI: 10.3847/2515-5172/ab158e