(Assuming space expands evenly) we don't notice the expansion of space inside objects because the electromagnetism and gravity are vastly more powerful on the scales we care about and nullifies the effect. That's why atoms and galaxies in the Local Group stay together, but everything farther away is flying further away.
If expansion never stops accelerating, one day in the far future it will overcome all the other fundamental forces and everything will be torn apart in a Big Rip.
If expansion never stops accelerating, one day in the far future it will overcome all the other fundamental forces and everything will be torn apart in a Big Rip.