Short answer
A Muslim must believe that God (﷾) created Adam directly, from clay, by His own command. God (﷾) breathed a soul into him, taught him the names of things, and told the angels to bow to him. His wife was created from him, as Muslim scholars understand the Qur'an. All people descend from the two of them. So Adam did not descend from some earlier creature. He is God's (﷾) own work, made on purpose and placed on earth with a trust, a responsibility from God (﷾). Beyond that, the Qur'an is not a biology textbook. It does not set out how every animal and plant came to be, and Muslim scholars hold different views about that. The real question is not how animals changed over time. It is whether man, and the world he lives in, were made on purpose. The Qur'an answers that both were.
Full explanation
Adam was made directly, by God's (﷾) command
The Qur'an says God (﷾) created Adam from dust and then said to him "Be", and he was. It describes God (﷾) telling the angels that He was about to create man from clay, shape him, and breathe into him of His spirit. God (﷾) speaks of Adam as one He created with His own hands. This is a deliberate act of creation, not a slow change from another animal.
“The similitude of Jesus before Allah is as that of Adam; He created him from dust, then said to him: "Be". And he was.”
“Behold! thy Lord said to the angels: "I am about to create man, from sounding clay from mud moulded into shape; "When I have fashioned him (in due proportion) and breathed into him of My spirit, fall ye down in obeisance unto him."”
“(Allah) said: "O Iblis! What prevents thee from prostrating thyself to one whom I have created with my hands? Art thou haughty? Or art thou one of the high (and mighty) ones?"”
Taught, honoured and given a task
God (﷾) announced to the angels that He would place a khalifa (a steward or successor) on the earth. He taught Adam the names of all things, then commanded the angels to bow down to him. Man also took on the Trust (a responsibility from God (﷾)) that the heavens, the earth and the mountains refused. No other creature is described this way.
“Behold, thy Lord said to the angels: "I will create a vicegerent on earth." They said: "Wilt Thou place therein one who will make mischief therein and shed blood?- whilst we do celebrate Thy praises and glorify Thy holy (name)?" He said: "I know what ye know not."”
“And He taught Adam the names of all things; then He placed them before the angels, and said: "Tell me the names of these if ye are right."”
“And behold, We said to the angels: "Bow down to Adam" and they bowed down. Not so Iblis: he refused and was haughty: He was of those who reject Faith.”
“We did indeed offer the Trust to the Heavens and the Earth and the Mountains; but they refused to undertake it, being afraid thereof: but man undertook it;- He was indeed unjust and foolish;-”
One pair, one human family
The Qur'an says all people were created from a single person, and that his mate was made from that person. The Yusuf Ali translation renders this "of like nature". But the Arabic word minha literally means "from it". Muslim scholars have read it as Eve being created from Adam. From the two of them came countless men and women. The Qur'an adds that mankind was made from a male and a female. It says the most honoured in God's (﷾) sight is the most righteous, not the best-born. The Prophet (ﷺ) said Adam was made from soil taken from the whole earth. That is why his children differ in colour and temper as the earth does.
“O mankind! reverence your Guardian-Lord, who created you from a single person, created, of like nature, His mate, and from them twain scattered (like seeds) countless men and women;- reverence Allah, through whom ye demand your mutual (rights), and (reverence) the wombs (That bore you): for Allah ever watches over you.”
“O mankind! We created you from a single (pair) of a male and a female, and made you into nations and tribes, that ye may know each other (not that ye may despise (each other). Verily the most honoured of you in the sight of Allah is (he who is) the most righteous of you. And Allah has full knowledge and is well acquainted (with all things).”
Abu Musa al-Ash'ari reports that the Prophet (ﷺ) said Allah created Adam from a handful of soil taken from the whole of the earth, and so the children of Adam vary as the earth does: some red, some white, some black and some in between, some gentle and some rough, some bad and some good.
What the Qur'an leaves open
Beyond the creation of Adam and his wife, the Qur'an does not give an account of how every other living thing came to be. It does not ask to be read as a biology textbook. Muslims have held a range of views on the rest of the living world. Some reject evolution entirely. Others accept it for other living things while holding that Adam was created directly. The point a Muslim must hold is the one the Qur'an states: the first man was made directly and on purpose.
Sets out the range of positions Muslims hold on evolution. They range from rejecting it entirely, to accepting it for other living things while holding that Adam was created directly, and beyond.
Evolution does not answer the question of God (﷾)
Evolution is an account of how living things change over time. It begins with life that already exists, on a planet that already exists, in a universe that already obeys laws. Charles Darwin, the nineteenth-century scientist who set out the theory, takes that first life as given in his own book. How the first living cell arose from non-living chemistry is still an open problem, with no accepted account. Even if that is one day solved, the point stands. Grant every detail of the theory: it cannot say why the universe exists, why it has laws, or whether man was made on purpose. Those are the questions that matter, and the Qur'an answers them.
Explains how species change over time by natural selection acting on variation. The theory starts from life already present: the closing paragraph speaks of life having been first breathed into a few forms or into one, and from the second edition says this was done by the Creator.
“Behold, thy Lord said to the angels: "I will create a vicegerent on earth." They said: "Wilt Thou place therein one who will make mischief therein and shed blood?- whilst we do celebrate Thy praises and glorify Thy holy (name)?" He said: "I know what ye know not."”
Perspectives
The Islamic perspective
God (﷾) created Adam directly from clay, gave him a soul and knowledge, and made his wife from him. He made all mankind from the two of them. On the rest of the living world the Qur'an gives no detailed account, and Muslims hold different views.
The Qur'an says God (﷾) created Adam from dust by the command "Be".
The Qur'an describes God (﷾) announcing Adam's creation from clay and breathing His spirit into him.
The Qur'an says God (﷾) taught Adam the names of all things and had the angels bow to him.
The Qur'an says all mankind comes from a single pair, a male and a female.
Eve was made from Adam, as Muslim scholars read the Arabic minha ("from it") in 4:1.
Adam was created directly and deliberately, not descended from an earlier creature.
Muslims hold different views about how other living things came to be, since the Qur'an does not set this out.
Evolution starts from life that already exists, so it cannot answer whether the universe and man were made on purpose.