We know the universe expands with the space expansion. This follows from the FLRW solution that nicely explains the observed Hubble flow. (Since any existing acceleration is extremely small, let's assume for simplicity it is zero.)
A uniform expansion of space does not apply any forces on bodies, so they move exactly the same way as they would in a non-expanding space. For example, if two bodies are at rest relative to each other, but space between them expands, then they would remain at rest relative to each other the same way as they would in a non-expanding space. In other words, the expansion of space cannot be detected by observing the motion of massive bodies (as long as they don't cross the cosmic horizon).
The expansion does affect the speed of light, as observed remotely. The expansion also creates a cosmic horizon, beyond which nothing can be detected. Have these effects ben observed? Are there any experimental results showing that space actually expands as opposed to the galaxies simply flying apart on inertia?