My Adult Female RES fell on the floor from the sink. Her shell has a very small crack and bled. The bleeding stop as I put medicine and did some first aid. I'm contemplating if I should still take her to the vet because we don't even have a vet for reptiles here. I really need an advice on how to take care of her at home. And what should proper care I need.
You should visit a Vet with her. Like trond hansen wrote, there could be internal injury you can not see, but which could kill your turtle.
After this you have to prevent your turtle falling out of the sink again!
Turtles can, whether or not you believe it, climb very well. They do so for example, if they walk miles and miles to mate.
The fence should therefore be sleek, without ledges giving a foothold. How tall it should be depends on the size of your biggest turtle (in case you have more than one). The biggest should not could reach the top edge of the fence, if it "stands upright" in front of it.
(You only have to search for "turtle climb" to find a lot of videos on all platforms, about turtles which climb netting wire fences or use techniques like mountaineers who climbs inside crevices.)