No, you do not. RCIA--the Rite of Christian Initiation for Adults--is for those who have never been initiated into the Catholic Church. You have received your basic initiation by virtue of your Catholic baptism. What you need now is to learn the Church's teachings (which you are already doing) and make your first confession and First Communion and to be confirmed.
Under canon law, you as an adult do not need to undergo formal instruction in order to make your first confession or First Communion; informal instruction is sufficient. In fact, "Sacred ministers may not deny the sacraments to whose who opportunely ask for them, are properly disposed, and are not prohibited by law from receiving them" (CIC 843).
You must know, understand, and accept the Church's teachings concerning confession and the Holy Eucharist. The person who teaches these to you does not need to be a priest or catechist, just someone who knows and will give you the straight story concerning the Church's teachings on these sacraments as found, for example, in the Catechism of the Catholic Church (see numbers 1322-1498 in the Catechism, about 40 pages of material to study).
Once you understand and accept these teachings, you can go to confession--anonymously if you wish--and explain to the priest that you were raised outside the Church and that this is your first confession. You will need to confess all the post-baptismal mortal sins that you can remember, indicating to the degree possible the number of times or the frequency with which these were committed. This needs to be done before going to First Communion.
Because you will have learned and accepted the Church's teaching concerning the Eucharist, you can begin going to Communion like any other Catholic. No special permission is necessary. "Any baptized person who is not forbidden by law may and must be admitted to Holy Communion" (CIC 912). Basically, those forbidden by law are children who are too young, uninstructed, or improperly disposed (CIC 913-4), people under a penalty of excommunication or interdict (CIC 915), those who have committed a mortal sin since their last confession (CIC 916), those who have already received the Eucharist that day (unless they take an active part in a second Mass or if special circumstances apply [CIC 918, 921:2]), and those who have not fasted for an hour before going to Communion (CIC 919).
To receive confirmation, which you should do as soon as possible (CIC 890-1), you will need more instruction: "Apart from the danger of death, to receive confirmation lawfully a person who has the use of reason must be suitably instructed, properly disposed, and able to renew the baptismal promises" (CIC 889:2). For this instruction your parish may put you in an RCIA class for the sake of convenience.
You do not need to receive confirmation before beginning to go to confession and beginning to receive the Eucharist. Nothing in canon law requires that, and you may begin practicing those parts of a Catholic's sacramental life as soon as you have understood and accepted the Church's teachings concerning them.