I know that in many cultures it's normal, for women, to cover up your whole body when you want to swim indoors, outdoors or just want to use the bathtub. What I'm wondering about is if there's anything in the Quran, hadiths or the different judical houses in Islam that says that's how it's supposed to be?
There is verse in Qur'an that talks about modesty
And tell the believing women to reduce [some] of their vision and guard their private parts and not expose their adornment except that which [necessarily] appears thereof and to wrap [a portion of] their headcovers over their chests and not expose their adornment except to their husbands, their fathers, their husbands' fathers, their sons, their husbands' sons, their brothers, their brothers' sons, their sisters' sons, their women, that which their right hands possess, or those male attendants having no physical desire, or children who are not yet aware of the private aspects of women. And let them not stamp their feet to make known what they conceal of their adornment. And turn to Allah in repentance, all of you, O believers, that you might succeed. An-Nūr 31
There is hadith that talks about going out with husband
Whenever the Prophet intended to proceed on a journey, he used to draw lots amongst his wives and would take the one upon whom the lot fell. Once, before setting out for Jihad, he drew lots amongst us and the lot came to me; so I went with the Prophet; and that happened after the revelation of the Verse Hijab (i.e. veiling). - Sahih al-Bukhari Book 52, Hadith 130
Not very detailed answer but you get the point :) and reference.
I did not bold list of mehram but that's understood.
Islam does not allow women to show her body while swimming or bathing. There are many ahadith and verses of Quran that talk about women modesty. As Prophet Muhammad (PBUH) said: "If a woman reaches maturity and she don't cover herself then her prayers will not be accepted." A verse from Qur'an: And if have no shame remaining, then whatever you do, Allah is no longer concerned with you. I am not good at English. But I just tried to give you a concept through authentic references. I hope you would understand these easily. But if she is in private with no other men or women she doesn't have to cover her body.