First of all, welcome! I also converted to Catholicism and I can say that the decision was the right one. I will pray for you and I hope that you can have the joy of being in full communion with the Catholic Church.
I don’t know much about the Oriental Orthodox Church but I think you may need to be baptised again due to the differences in how the two faiths view Christ. It also depends on the formula under which you were baptised. You will need to speak to your Priest, he will tell you how soon you could convert. That will usually depend on how well you understand the Catholic Faith and how long you have been attending Mass. If you want to convert as soon as possible, then talk it over with the Priest. There are no fixed rules, the Priest who received me into the Church had an amazing experience with a man in hospital, I’ve put the story at the end of this answer.
Converting involves the Priest making sure that you understand and believe the teachings of the Catholic Church. You then need to go through your formal reception (conversion) into the Catholic Church.
Your reception into the Catholic Church takes place during Mass, usually at Sunday Mass. If you are not going to be baptised, you will need to go to confession the night before, or you might be able to arrange to have your confession just before the service. Whenever your confession takes place, you will receive absolution from the Priest.
The part of the Mass where you are received into the Church takes place between the Readings (the Liturgy of the Word) and the start of the Liturgy of the Eucharist, at the part where we normally say the Nicene Creed on Sundays… You will need to make a profession of faith (
not a confession as one of the people converting along with me thought!

) in front of the congregation. After you make the profession of faith, you would then be baptised and then Confirmed. Obviously, if you are baptised, you do not need to get absolution from the Priest, as your sins will be washed away by the baptism. If you do not need to be baptised, you would be Confirmed immediately after your profession of faith. You will then receive your First Communion during the Mass.
The Oriental Orthodox Churches have teachings about Christ that are heretical to the Catholic Church’s teaching. I know that there are many Eastern Catholic Churches which are in full communion with Rome and therefore accept the primacy of the Pope.
I apologise if this doesn’t make sense, I am waiting for my father to call and tell me that my sister has had her baby. Could you please say a prayer for her? She has been in labor for almost two days and is very tired and in a lot of pain. The midwife said that she might give birth before midnight and it is 11.30 pm here!
I only realised that you’d had several answers after posting this, there was only one when I started typing. It took me an hour due to being so distracted!
Father was covering for the chaplin of a large hospital, who was on holiday, and was asked to see a man who was begging for a Catholic Priest. Thinking that this might be someone with something they needed to confess, Father went to see him right away. Instead he found that the man wasn’t a Catholic but wanted to become one immediately, despite not knowing very much about the Faith. Father explained that he would need to learn the basics before he could convert because he would need to make a profession of faith. Father talked with the man for an hour, explaining about the Real Presence in the Blessed Sacrament and a couple of other basics of the Faith. He gave the man a book that explained what Catholics believe and told the man he would be back to see him the next day. When he saw him, he found that the man had read the whole book and was once again begging to become a Catholic immediately. As the man was not in danger of death (he wasn’t seriously ill either), Father explained that to go ahead, he’d need to get something that proved he’d been baptised and a couple of other things like his birth certificate.
That night about 3am, Father got a phone call from the nurse on the ward, asking if Father would come to see this man, because he had been crying and begging them to get the priest immediately. Father drove to the hospital, and out of habit, took the case he had with consecrated hosts, the oils of chrism etc. When Father arrived, the man was in tears, begging him to make him a Catholic. Father told me that he didn’t know what made him agree, but he baptised, confirmed and gave the man his First Communion. The man was so happy, he was almost glowing and he thanked Father and apologised for waking him. Father drove home, which took about 25 minutes. As he got in the door, the phone rang. It was the nurse on the ward again. The man had died less than five minutes after Father left.

The cause of death wasn’t something that could be due to the man’s distress either.