Don't use baby shampoo or ANY shampoo that people use, since it can cause irritation and discomfort for your rat. Gently stroke the rat's tail with the damp toothbrush, and brush AWAY from the rat's body. If you try to brush towards the body, you can harm the delicate skin structure that makes your rat's tail look like a rat tail. You can also harm the tail's skin if you brush too hard. Your pet won't appreciate it, and you won't appreciate the ensuing vet bill.
Some people have their rat's nails clipped.
If you decide this is necessary, please go to a vet experienced with rat anatomy, or find a well-versed rat owner to do this for you. If you decide to do this yourself anyway, keep a styptic pencil or styptic powder ready, in case you accidentally nick or cut the little vein that runs inside the rat's toenails by cutting too much of the nail off, slipping with the clipping instrument, or having the rat get excited about his maicure and jaming his toes into the clipper the wrong way.