Hazrat Muhammad said that when you see the moon of the month of Rajjb, recite the following prayer: اَللّٰہُمَّ بَارِکْ لَنَا فِیْ رَجَبَ وَ شَعْبَانَ وَ بَلِّغْنَا رَمَضَانَ (رواہ البیہقی)
Nafl Namaz: On the first night of Rajjb, after Namaz e Isha, offer offer 10 rakat Nafl Namaz with 5 salaams, recite Surah Kaafroon, and Surah Ikhlas 3 times after Surah Fatihah. Insha Allah, the one who offers these Nafl would amongst the martyrs when he dies and his rank would increase 1000 times.
First night: After Namaz isha, offer 4 rakat Nafl with two Salam. After Surah Fatihah, recite Surah Alm Nashrah, Surah ikhlas, Surah falaq and Surah Nas once. After Salam, recite Qalma Tauhid and Durud Shareef 33 times. Repeat this for the next two rakat too. When done, pray for the fulfilment of any of your needs and Allah's blessings.
For all Friday nights (according to the Islamic Day) in the month of Rajab, offer two rakat Nafl after Isha Namaz and in the first rakat, recite the last Ruquh of Surah Baqarah (ٰامَنَ الرَّسُوْلُ --- کَافِرِیْن) 7 times. In the second rakat, recite the last verse of Surah Hashr 7 times (ہُوَ اللّٰہُ الَّذِیْ ---- اَلْحَکِیْمُ) and after salaam, make a dua for any of your need.
On the first Friday of this month, after Maghrib prayer, offer 12 rakat nafl with 6 Salam. Recite Surah Qadr 3 times and Surah Ikhlas 12 times and after that 70 times, recite the following verse اَللّٰہُمَّ صَلِّ عَلٰی مُحَمَّدِ نِ النَّبِیْ الامُیِّ وَ عَلٰٓی ٰالِہٖ وَسَلِّمْ تَسْلِیْماَ and after that, in the sajda or prostration, recite this سُبُّحٌ قُدُّوْسٌ رَبُّ المَلٓائِکَۃِ وَالرُّوْحِ 70 times and after getting up from the Sajda, recite this رَبِّ اغْفِرْ وَارْحَمْ وَ تَجَاوَزْ عَمَّا تَعْلَمُ اِنَّکَ اَنْتَ الْعَزِیْزُ الْاَعْظَمْ 70 times followed by another Sajda like the one before. Make a dua after this and Allah will bless you Insha’Allah.
On any night in the month of Rajab pray 12 rakat namaz with three Salams. In the first four rakat after reciting surah Fatiha recite surah Qadr three times in each rakat. After Salam recite لَآ اِلٰہَ اِلاَّ اللّٰہُ الْمَلِکُ الْحَقُّ الْمُبِیْنُ 70 times. In the second four rakats after reciting surah Fatiha recite surah Nasar three times in each rakat. After Salam recite اِنَّکَ قَوِیٌّ مُّعِیْنٌ وَّاحِدِیٌّ دَلِیْلٌ بِحَقِّ اِیَّاکَ نَعْبُدُ وَاِیَّاکَ نَسْتَعِیْنُ 70 times. In the third four rakats after reciting surah Fatiha recite surah Ikhlas three times in each rakat. After saying the Salam recite surah Alam Nashrah 70 times. Then pray to Allah Almighty. Inshallah whatever you wish for will be granted. In the month of Rajab-ul-Marajab on the 15th after any prayer the following Istagfar must be recited once. Allah Almighty will erase all the person's sins and convert them into good deeds.
اَسْتَغْفِرُ اللّٰہَ الَّذِیْ لَآاِلٰہَ اِلاَّ ہُوَ الْحَیُّ الْقَیُّوْمُ غَفَّارُ الذُّنُوْبِ وَسَتَّارُ الْعُیُوْبِ وَاَتُوْبُ اِلَیْہِ In the month of Rajab on any day after namaz Zuhr, Maghrib or Isha recite surah Kahf once, surah Yaseen once, surah Ha Mim once, surah Dukhan once and surah Ma'arij once. After this recite surah Ikhlas 1000 times.