Could you inform me if there is a particular Du'a which could be recited when moving in a new house, apartment, etc.? If there is such a Du'a could you bring an evidence, please? If there is no such a Du'a can I just recite some verses from Surah al-Baqarah? What's the best to do when moving in?

Praise be to Allah, the Lord of the World; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

There is no sound evidence proving that the person should supplicate Allah with a particular Du'a or slaughter a beast or do any other special form of worship when entering a new house.
However, if the person slaughters or gives some money for poor people as an act of gratitude to Allah for providing him with a new house, this is good. It is likeable for the person entering a new house to say: "I seek refuge in the Perfect Word of Allah from the evil of what He has created" (أعوذ بكلمات التامات من شر ما خلق) as reported by Imam Muslim from Khawla Bint Hakim who said: "I heard Allah's Messenger (may peace be upon him) as saying: When anyone lands at a place, and then says: "I seek refuge in the Perfect Word of Allah from the evil of what He has created", nothing would harm him until he marches from that stopping place" .
There is no difference between this and a house that he will live in permanently, a rented one or even a "maison de passage".
It is recommendable for him also to recite some Qur'an especially Surah Al-Baqarah to chase the devils away from his house.
Reported Muslim that the Prophet (Sallallahu Alaihi wa Sallam) said: "Offer some of your prayers (Nawafil) at home, and do not take your houses as graves. In fact, Satan flees from the house where Surah Al-Baqarah is recited" .
We advise the questioner also to adhere to the known supplications (Du'a) of entering one's home and going out of it.
Allah knows best.