1. Creation of Adam (Quran 38:71-72)
Allah speaks of Adam's creation from clay:

Arabic:
‏إِذْ قَالَ رَبُّكَ لِلْمَلَائِكَةِ إِنِّي خَالِقٌ بَشَرًا مِّن طِينٍ ‎
‏فَإِذَا سَوَّيْتُهُ وَنَفَخْتُ فِيهِ مِن رُّوحِي فَقَعُوا لَهُ سَاجِدِينَ ‎
Translation:
"When your Lord said to the angels, 'Indeed, I am going to create a human being from clay. So when I have proportioned him and breathed into him of My spirit, then fall down to him in prostration.'" (Surah Sad 38:71-72)

2. Teaching Adam Knowledge (Quran 2:31-33)

Allah teaches Adam the names of all things:

Arabic:
وَعَلَّمَ آدَمَ الْأَسْمَاءَ كُلَّهَا ثُمَّ عَرَضَهُمْ عَلَى الْمَلَائِكَةِ فَقَالَ أَنبِئُونِي بِأَسْمَاءِ هَـٰؤُلَاءِ إِن كُنتُمْ صَادِقِينَ
قَالُوا سُبْحَانَكَ لَا عِلْمَ لَنَا إِلَّا مَا عَلَّمْتَنَا ۖ إِنَّكَ أَنتَ الْعَلِيمُ الْحَكِيمُ
قَالَ يَا آدَمُ أَنبِئْهُم بِأَسْمَائِهِمْ ۖ فَلَمَّا أَنبَأَهُم بِأَسْمَائِهِمْ قَالَ أَلَمْ أَقُل لَّكُمْ إِنِّي أَعْلَمُ غَيْبَ السَّمَاوَاتِ وَالْأَرْضِ وَأَعْلَمُ مَا تُبْدُونَ وَمَا كُنتُمْ تَكْتُمُونَ
Translation:
"And He taught Adam the names—all of them. Then He showed them to the angels and said, 'Inform Me of the names of these, if you are truthful.' They said, 'Exalted are You; we have no knowledge except what You have taught us. Indeed, it is You who is the Knowing, the Wise.' He said, 'O Adam, inform them of their names.' And when he had informed them of their names, He said, 'Did I not tell you that I know the unseen [aspects] of the heavens and the earth? And I know what you reveal and what you have concealed.'" (Surah Al-Baqarah 2:31-33)

3. Prostration of the Angels (Quran 2:34)
The angels prostrate to Adam except Iblis:

Arabic:
‏وَإِذْ قُلْنَا لِلْمَلَائِكَةِ اسْجُدُوا لِآدَمَ فَسَجَدُوا إِلَّا إِبْلِيسَ أَبَىٰ وَاسْتَكْبَرَ وَكَانَ مِنَ الْكَافِرِينَ
Translation:
"And [mention] when We said to the angels, 'Prostrate before Adam'; so they prostrated, except for Iblis. He refused and was arrogant and became of the disbelievers."
(Surah Al-Baqarah 2:34)

4. The Temptation and Fall (Quran 7:19-22)
The temptation of Adam and his wife:

Arabic:
وَيَا آدَمُ اسْكُنْ أَنتَ وَزَوْجُكَ الْجَنَّةَ فَكُلَا مِنْ حَيْثُ شِئْتُمَا وَلَا تَقْرَبَا هَـٰذِهِ الشَّجَرَةَ فَتَكُونَا مِنَ الظَّالِمِينَ
فَوَسْوَسَ لَهُمَا الشَّيْطَانُ لِيُبْدِيَ لَهُمَا مَا وُورِيَ عَنْهُمَا مِن سَوْآتِهِمَا وَقَالَ مَا نَهَاكُمَا رَبُّكُمَا عَنْ هَـٰذِهِ الشَّجَرَةِ إِلَّا أَن تَكُونَا مَلَكَيْنِ أَوْ تَكُونَا مِنَ الْخَالِدِينَ
وَقَاسَمَهُمَا إِنِّي لَكُمَا لَمِنَ النَّاصِحِينَ
فَدَلَّاهُمَا بِغُرُورٍ ۚ فَلَمَّا ذَاقَا الشَّجَرَةَ بَدَتْ لَهُمَا سَوْآتُهُمَا وَطَفِقَا يَخْصِفَانِ عَلَيْهِمَا مِن وَرَقِ الْجَنَّةِ ۖ وَنَادَاهُمَا رَبُّهُمَا أَلَمْ أَنْهَكُمَا عَن تِلْكُمَا الشَّجَرَةِ وَأَقُل لَّكُمَا إِنَّ الشَّيْطَانَ لَكُمَا عَدُوٌّ مُّبِينٌ

Translation:
"And, 'O Adam, dwell, you and your wife, in Paradise and eat from wherever you will, but do not approach this tree, lest you be among the wrongdoers.' But Satan whispered to them to make apparent to them that which was concealed from them of their private parts. He said, 'Your Lord did not forbid you this tree except that you become angels or become of the immortal.' And he swore [by Allah] to them, 'Indeed, I am to you from among the sincere advisors.' So he made them fall, through deception. And when they tasted of the tree, their private parts became apparent to them, and they began to fasten together over themselves from the leaves of Paradise. And their Lord called to them, 'Did I not forbid you from that tree and tell you that Satan is to you a clear enemy?'"
(Surah Al-A'raf 7:19-22)

5. Repentance and Forgiveness (Quran 2:37)
Adam repents, and Allah forgives him

Arabic:
‏فَتَلَقَّىٰ آدَمُ مِن رَّبِّهِۦ كَلِمَـٰتٍۢ فَتَابَ عَلَيْهِ ۚ إِنَّهُۥ هُوَ ٱلتَّوَّابُ ٱلرَّحِيمُ
Translation:
"Then Adam received from his Lord [some] words, and He accepted his repentance. Indeed, it is He who is the Accepting of repentance, the Merciful."
(Surah Al-Baqarah 2:37)

These verses encapsulate the story of Adam in the Quran, emphasizing his creation, the knowledge given to him, the trial with Iblis, his subsequent fall, and the mercy of Allah.