Short answer
Nuh (Noah) was the first messenger (rasul) sent to the people of the earth, as the Prophet ﷺ said in the hadith of intercession. Adam was the first human and the first prophet (nabi). Muhammad ﷺ is the last of both.
Sunni creed holds that the first prophet was Adam and the last is Muhammad ﷺ, and that believing in all of them is fard without fixing their number. Hanafi theologians commonly define a rasul as a prophet sent with a sacred law or book to deliver, and a nabi as broader: every rasul is a nabi, but not every nabi is a rasul.
The four schools agree on this.
Explanation
In the long hadith of intercession, people on the Day of Judgement go from prophet to prophet. When they come to Nuh they say, "O Nuh, you are the first of the messengers to the people of the earth" (Sahih al-Bukhari). The Quran also begins its list of revelation with him: "We have revealed to you as We revealed to Noah and the prophets after him" (4:163).
Why Nuh and not Adam? Adam was a prophet: Allah taught him and spoke to him (2:31, 2:37), and a hadith reported by Ibn Hibban and others calls him a prophet who was spoken to. But in Adam's time people were on one belief, so he guided his children rather than being sent to a people who had gone astray. Many scholars explain that the first idol worship appeared among the people of Nuh, so he was the first sent to call a nation back from shirk.
Allah says He "chose Adam and Noah and the family of Abraham and the family of Imran over the worlds" (3:33). Nuh is also one of the five messengers of firm resolve, with Ibrahim, Musa, Isa and Muhammad ﷺ. His story of patience over many centuries is in the story of Prophet Nuh.
How many prophets and messengers were there? Only Allah knows the full count; 25 are named in the Quran. See How many prophets are there in Islam? and the full list on Stories of the Prophets.
The evidence
Quran, Surah An-Nisaa 4:163
۞ إِنَّآ أَوْحَيْنَآ إِلَيْكَ كَمَآ أَوْحَيْنَآ إِلَىٰ نُوحٍۢ وَٱلنَّبِيِّۦنَ مِنۢ بَعْدِهِۦ ۚ وَأَوْحَيْنَآ إِلَىٰٓ إِبْرَٰهِيمَ وَإِسْمَٰعِيلَ وَإِسْحَٰقَ وَيَعْقُوبَ وَٱلْأَسْبَاطِ وَعِيسَىٰ وَأَيُّوبَ وَيُونُسَ وَهَٰرُونَ وَسُلَيْمَٰنَ ۚ وَءَاتَيْنَا دَاوُۥدَ زَبُورًۭا ﴿١٦٣﴾
Indeed, We have revealed to you, [O Muhammad], as We revealed to Noah and the prophets after him. And we revealed to Abraham, Ishmael, Isaac, Jacob, the Descendants, Jesus, Job, Jonah, Aaron, and Solomon, and to David We gave the book [of Psalms].
Revelation to the messengers is listed starting with Nuh.
Quran, Surah Aal-i-Imraan 3:33
۞ إِنَّ ٱللَّهَ ٱصْطَفَىٰٓ ءَادَمَ وَنُوحًۭا وَءَالَ إِبْرَٰهِيمَ وَءَالَ عِمْرَٰنَ عَلَى ٱلْعَٰلَمِينَ ﴿٣٣﴾
Indeed, Allah chose Adam and Noah and the family of Abraham and the family of 'Imran over the worlds -
Adam and Nuh are among those Allah chose.
Sahih al-Bukhari 4712
Sahih... يا نوح إنك أنت أول الرسل إلى أهل الأرض ...
"O Nuh, you are the first of the messengers to the people of the earth."
States that Nuh was the first messenger.
Where the ruling is recorded
- Sharh al-Aqa'id al-Nasafiyya (al-Taftazani), Section on the prophets. States that the first prophet was Adam and the last Muhammad ﷺ, and discusses rasul and nabi.
- Tafsir Ibn Kathir, Commentary on Surah al-Nisa, verse 163. Discusses the order of the prophets and messengers.
Checked by our team. For a more accurate answer to your own situation, consult a scholar. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.