All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
Allah has chosen the month of Ramadan among all months by making it the month where the good deeds are given double reward, above all fasting and praying Qiyam at night. Allah has given the last nights of Ramadan greater importance by setting Laylat Al Qadr (the Night of Decree) among them. This is why the Prophet used to do as much acts of worship as possible in these last ten nights of Ramadan.
He used to worship Allah in these nights more than any other nights. He used to do I'tikaf with his wives during these last days of Ramadan. It is desirable for a Muslim to perform as much good deeds as he can in Ramadan (praying, giving alms…). He should remember Allah as much as he can and should recite the Quran as much as he can. On the other hand, we do not know anything peculiar to the first ten days or the second ten days of Ramadan.
Allah knows best.