Imam mentions two important activities of Prophet in the month of Shaban; fasting and engaging in worships.
About fasting in this month, there are many Ahadith that encourage the wakeful heart. About the fast in the Shaban Imam Reza has said,

عيون أخبار الرضا عليه السلام و الخصال بسندهِ عَنِ الْعَبَّاسِ بْنِ هِلَالٍ قَالَ سَمِعْتُ أَبَا الْحَسَنِ عَلِيَّ بْنَ مُوسَى الرِّضَا ع يَقُولُ مَنْ صَامَ مِنْ شَعْبَانَ يَوْماً وَاحِداً ابْتِغَاءَ ثَوَابِ اللَّهِ دَخَلَ الْجَنَّةَ وَ مَنِ اسْتَغْفَرَ اللَّهَ فِي كُلِّ يَوْمٍ مِنْ شَعْبَانَ سَبْعِينَ مَرَّةً حُشِرَ يَوْمَ الْقِيَامَةِ فِي زُمْرَةِ رَسُولِ اللَّهِ ص وَ وَجَبَتْ لَهُ مِنَ اللَّهِ الْكَرَامَةُ وَ مَنْ تَصَدَّقَ فِي شَعْبَانَ بِصَدَقَةٍ وَ لَوْ بِشِقِّ تَمْرَةٍ حَرَّمَ اللَّهُ جَسَدَهُ عَلَى النَّارِ وَ مَنْ صَامَ ثَلَاثَةَ أَيَّامٍ مِنْ شَعْبَانَ وَ وَصَلَهَا بِصِيَامِ شَهْرِ رَمَضَانَ كَتَبَ اللَّهُ لَهُ صَوْمَ شَهْرَيْنِ مُتَتَابِعَيْنِ

“Whoever fasts one day of Shaban, just for the sake of God’s reward he will enter the heaven and whoever repents to Allah seventy times, on the Day of Judgment he will be raised with the Prophet and God’s appreciation will be obligatory on him. Whoever donates in this month even by half of a date, God will forbid his body to go to hell. Whoever fasts three days of Shaban and connects these days to the month of Ramadan, God will give him the reward for the two complete months”[1]. Also Imam Sadeq

ثواب الأعمال بسنده عن ُ أَبَا عَبْدِ اللَّهِ ع يَقُولُ مَنْ صَامَ أَوَّلَ يَوْمٍ مِنْ شَعْبَانَ وَجَبَتْ لَهُ الْجَنَّةُ بَتَّةً وَ مَنْ صَامَ يَوْمَيْنِ نَظَرَ اللَّهُ إِلَيْهِ فِي كُلِّ يَوْمٍ وَ لَيْلَةٍ فِي دَارِ الدُّنْيَا وَ دَامَ نَظَرُهُ إِلَيْهِ فِي الْجَنَّةِ وَ مَنْ صَامَ ثَلَاثَةَ أَيَّامٍ زَارَ اللَّهَ فِي عَرْشِهِ مِنْ جَنَّتِهِ فِي كُلِّ يَوْم

 encourages us by a hadith that says, “Anyone who fasts the first day of Shaban, it is obligated upon him to go to heaven and anyone who fasts two days of Shaban, God will look at him every day and night in this world and he will continuously be under the observance of God’s mercy. Whoever fast three days he will visit God in the heaven in Allah’s throne”[2].
Also in another Hadith Imam Sadeq is not just encouraging the narrator, but also told him to reassure his friends and family to fast in the month of Shaban.

الْإِقْبَالُ، وَ مَجَالِسُ الشَّيْخِ، بِإِسْنَادِهِمَا عَنْ صَفْوَانَ الْجَمَّالِ قَالَ قَالَ لِي أَبُو عَبْدِ اللَّهِ ع حُثَّ مَنْ فِي نَاحِيَتِكَ عَلَى صَوْمِ شَعْبَانَ فَقُلْتُ جُعِلْتُ فِدَاكَ تَرَى فِيهَا شَيْئاً فَقَالَ نَعَمْ إِنَّ رَسُولَ اللَّهِ ص كَانَ إِذَا رَأَى هِلَالَ شَعْبَانَ أَمَرَ مُنَادِياً يُنَادِي فِي الْمَدِينَةِ يَا أَهْلَ يَثْرِبَ إِنِّي رَسُولُ رَسُولِ اللَّهِ إِلَيْكُمْ أَلَا إِنَّ شَعْبَانَ شَهْرِي فَرَحِمَ اللَّهُ مَنْ أَعَانَنِي عَلَى شَهْرِي ثُمَّ قَالَ إِنَّ أَمِيرَ الْمُؤْمِنِينَ ع كَانَ يَقُولُ مَا فَاتَنِي صَوْمُ شَعْبَانَ مُنْذُ سَمِعْتُ مُنَادِيَ رَسُولِ اللَّهِ ص يُنَادِي فِي شَعْبَانَ فَلَنْ تَفُوتَنِي أَيَّامَ حَيَاتِي صَوْمُ شَعْبَانَ إِنْ شَاءَ اللَّهُ ثُمَّ كَانَ ع يَقُولُ صَوْمُ شَهْرَيْنِ مُتَتابِعَيْنِ تَوْبَةٌ مِنَ اللَّهِ

Safvan Jammal narrates that, “Imam told me to encourage whoever is around you.”I asked, “Is there any (special) reward for fasting in Shaban”. Imam replied, “Yes, whenever the Prophet saw the moon (that shows the starting) of Shaban he ordered somebody to call all the people, and he would say: O the people of Yathreb, I am the messenger of Allah to you. Be aware Shaban is my month and may Allah bless those who help me in this month” Then Imam Sadeq said, “The Commander of the Faithful has said, “From the moment I heard this, I have never missed fasting in the month of Shaban and I will not miss it InshAllah.” Then Imam Ali said, “Fasting two month continuously is the repentance to Allah.”[3]
We must also request God for success and along with Imam Sajjad we must say,
“This is the month of Thy Prophet, the Messenger-in-chief, the month of Sha'ban surrounded by Thy mercy and blessings from all sides,   In which the Messenger of Allah (peace and blessings of Allah be on him and on his children) used to exert himself willingly in the matter of fasting and praying in its days and nights, to show his humility before Thee for the respect and honor Thou bestowed on him, until his last breath.”

Notes
[1] بحارالانوار  ج 94ص  71
[2] Same address
[3] Same address