Is sexual intercourse with one's wife considered adultery? Is it permissible to take a bath along with one's wife?
Please advise.

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.

Sexual intercourse with one's wife is, by no means, adultery. On the contrary, it is a great act of worship by means of which one comes close to Allaah Almighty. The Messenger of Allaah, sallallaahu ‘alayhi wa sallam, said: "…and in man's sexual intercourse (with his wife) there is a charity." They (the Companions) said, "O Messenger of Allaah! Is there a reward for anyone among us who satisfies his sexual desire (with his wife)?" He said, "Tell me, if he fulfilled it unlawfully, would it not be regarded as a sin committed by him? Similarly, if he fulfilled it lawfully, he would receive a reward (for it)." [Muslim on the authority of Abu Tharr, May Allaah Be Pleased with him]

Adultery, on the other hand, is when a man has sexual intercourse with a woman other than his wife. It is one of the grave major sins, as confirmed by Allaah Almighty Who Says (what means): {And do not approach unlawful sexual intercourse. Indeed, it is ever an immorality and is evil as a way.}[Quran 17:32] At the same time, Allaah Praised the faithful believers as they keep themselves away from it (adultery), Saying (what means): {And they who guard their private parts, except from their wives or those their right hands possess, for indeed, they will not be blamed.}[Quran 23:5-6]

Sexual intercourse with one's wife is as different from adultery as much as heaven is from the earth. Sexual intercourse with one's wife results in lowering the gaze, maintaining one's chastity, guarding the private parts (genitals) and the proliferation of the people of the Ummah, as stated by the Prophet, sallallaahu ‘alayhi wa sallam:

Marry so that you would proliferate, for I am going to take pride in you (because of your large number) before the other nations on the Day of Judgment. [‘Abdul-Razzaaq]

This is in addition to the other great benefits of marriage.

Adultery, on the contrary, brings about a lot of evil things in the World and the Hereafter, incurs the Wrath of Allaah, the Creator, the Almighty, and leads to loss of family, mixing of lineages, violating honor, and increasing the number of illegitimate children born from adultery, who find neither a maintainer nor a refuge; thereupon they become a source of malice against their communities, as seen in those communities in which this vice has become widespread…to the end of the series of the demerits of adultery.

For this reason, Allaah Almighty Joined adultery with Shirk and killing souls (unlawfully), as stated in His Saying (what means): {And those who do not invoke with Allaah another deity or kill the soul which Allaah Has Forbidden [to be killed], except by right, and do not commit unlawful sexual intercourse. And whoever should do that will meet a penalty. Multiplied for him is the punishment on the Day of Resurrection, and he will abide therein humiliated – except for those who repent, believe and do righteous work. For them Allaah Will Replace their evil deeds with good. And ever Is Allaah Forgiving and Merciful.}[Quran 25:68-70]

To take a bath along with one's wife is permissible due to what is proven on the authority of the Mother of the Believers, ‘Aa’ishah who said:

I used to make Ghusl with the Prophet, sallallaahu ‘alayhi wa sallam, from a single pot, in which our hands would go after each other in turn. [Al-Bukhari and Muslim]