To dress = to put on clothes.
e.g. A baby can not dress themselves./I dress myself every morning.
Dressing up would be used if you �re wearing a costume:
e.g. I am going to a fancy dress party next week, I am dressing up as a clown.
Every Halloween I dress up as a ghost.
PS. If I was going somewhere posh/fancy/wearing something smart, I would also use dressing up but I don �t know if that �s a colloquialism.
e.g. I �m going on a date tonight so I have to dress up!