Are there directions in space? After leaving our solar system what becomes?

Once you leave Earth, say, in the Solar System, you can use the Sun as a central reference, and the common direction of revolution of the planets as a secondary.

'Inward' is toward the Sun, 'Outward' is away from the Sun

'Spinwise' is moving in the general direction of planet revolution, 'Anti-Spinwise is in the opposite direction.

'Spin-Up' is going 'upwards' from the Sun-ecliptic plane, based on the Right-Hand Rule (curl the fingers of your right hand in the direction of Spin, and 'Up' is the direction your thumb will point.

'Spin-Down' is the opposite direction, of course.

Outside the Solar system, one could use a common Milky Way Galaxy version of the same thing.

Beyond the MW galaxy... well, I'll leave that to the geniuses that manage to develop inter-galactic travel....

As long as there is 3-dimensions of space and 1 dimension of time, you would have a coordinate system.

They navigate like many did before. They use the stars, since they are so far away and are a good stationary (in your view) reference point

Only that the universe expands radially.