It is narrated from Sayyiduna Ibn Umar (may Allah be pleased with them both) that when Sayyiduna Umar (may Allah be pleased with him) accepted Islam, people gathered around him and began saying: "Umar has left his religion!" I was on the roof of the house. Al-As bin Wa'il came, wearing a cloak of silk, and said: "Umar has left his religion, but I have given him my protection." Sayyiduna Ibn Umar (may Allah be pleased with them both) said: The people dispersed, and I was greatly amazed at his standing that day.
Sayyiduna Umar bin al-Khattab (may Allah be pleased with him) accepted Islam while Sayyiduna Abdullah bin Umar (may Allah be pleased with them both) was still a boy, so he too became Muslim through Sayyiduna Umar (may Allah be pleased with him). This is based on the hadith that is established from the Prophet (peace and blessings be upon him): Sayyiduna Ibn Umar (may Allah be pleased with them both) said: The Messenger of Allah (peace and blessings be upon him) reviewed me on the day of Uhud when I was fourteen (14) years old, and he (peace and blessings be upon him) considered me too young. It has also been said that Sayyidah Hafsah (may Allah be pleased with her) and Sayyiduna Abdullah (may Allah be pleased with him) both became Muslim before their father, and Abdullah was young at that time, so his Islam was completed through his father's Islam.
As for Sayyiduna al-Abbas bin Abd al-Muttalib (may Allah be pleased with him), he went out to Badr with the polytheists and was taken captive, until he ransomed himself and accepted Islam.
Hadith Referenceالسنن الكبرى للبيهقي / كتاب اللقطة / 12149