When I empty my bath the water back up in to the sink

There's a clog in the tub drain, very common. Buy some liquid drain cleaner and a hair trap. (There's still old fashioned lye crystals on the market too, but be careful none of them touch the tub, just in the drain, followed by water.)

Hi guys:

Chemicals are a poor solution to issues like this as you have two fixtures blocked which means there is a blockage down the line, past the sink drain and the only way to really resolve this long term will be to snake/cable the common drain line.

Here, I recommend that you rent a small hand-held electric snake and either snake the drain down the tub waste overflow (see image) or remove the PTRAP at the sink and snake down the sink drain.

If you go down the tub waste overflow you can expect resistance at about 18" in... this is the tub PTRAP... just push and spin at the same time and you will get past it.

If you go down the sink drain see if the snake comes with a "drop head" (see images) in the kit and use that as this will increase the chances that the snake will go down the drain and not up the sink vent, OK?

Finally, be sure to wear goggles and a pair of leather gloves for safety sake, OK?

Good luck!

Mark