Hi Bear Claw,
Personally, I find it very interesting that you are interested in such a matter.
For someone today, outside of Christianity, looking at Christianity can be bizzare considering the state of things. It would appear that there are tons of people claiming Christianity, but not in agreement with one another.
Currently, there are 3 Types of Christianity in the world today: Catholic, Orthodox & Protestant. There are a handful of divisions among the Orthodox (Coptic vs. Oriental vs. Orthodox vs. Old Calendar) and thousands of divisions among the Protestants (Baptist, Evangelical, Lutheran, Anglican, Non-Denominational, 7th Day Adventists, etc.). Although there are various Rites within the Catholic Church, I think about 12, there is complete unity within the Catholic Church. This makes the Catholic Church stand out a bit from the crowds within Christianity. The Pope is the unifying figure among all Rites of the Catholic Church.
Your question though is between 2 of the 3 Types: Catholic & Orthodox.
Both Churches claim their Church was started by Jesus in 29 or 33 ad or there abouts. Both claim Apostolic Succession. In both of these claims, they are correct. Originally all the current Orthodox Patriarchs as successors of the Apostles in full communion with the Pope of the Church of Rome.
In 1054ad the two Churches officially split from one another with mutual ex-communications, the Coptic/Oriental Orthodox had split off earlier. This was due to language barriers, which made discussing the Faith difficult because the translations weren’t always the best and misunderstandings arose on both sides. It was also due to Political situations in the East vs. the West. Lastly, and I would propose based on my studies that this was soley due to the language barriers of the time, there were doctrinal differences. For example, Orthodox pray for the dead and believe in God’s Mercy, but they do not believe in what Catholics have named Purgatory because at the Council of Florence, they thought that the Romans were speaking of a 2nd “Temporary” Fire when speaking of Purgatory, but they knew that God is Fire and that there is only 1 Eternal Fire - due to bad translators/translations they did not understand that it is the soul that experiences the “purifying” affects of that 1 Eternal Fire temporarily as once that soul is completely purified, the affects to that soul of the same 1 Eternal Fire is Illumination/Deification. The mutual ex-communications of the Greek Orthodox Church and the Catholic Church were mutually lifted in the 1960s.
After the split, the Church which remained intact was the Catholic Church which has no divisions among its members as it has the Pope as the unifying force among Catholic Patriarchs and Bishops. The Orthodox Church, as mentioned before has divisions among its members with some Patriarchs united, but not all, and whose who are united within the Orthodox Church has changed several times through out the centuries.
It is the Catholic Church which over time has maintained the Fullness of Faith; however, that being said, the Catholic Church is clear in that the Orthodox Churches continue to maintain valid Sacraments/Mysteries: Baptism, Chrismation/Confirmation, Holy Orders, Holy Eucharist, Marriage, Annointing of the Sick/Last Rites.
As an Atheist, should you decide to become a Christian you would do well to become Baptized into a Protestant Christian Church as most Protestant Churches do provide valid Baptisms and with a valid Baptism you can become a true Christian.
You would do better to become an Orthodox Christian as in addition to a valid Baptism, you would also be able to receive the God of the Universe in the Holy Eucharist and other Sacraments/Mysteries.
Yet you would do even better to become a Catholic, because you would receive all of those Sacraments as well as have the assurance that when you are taught, you are taught from the Fullness of Faith with nothing left out being led by the Pope who has been promised to us by Christ to never error in matters of faith and doctrine.