How to Become Catholic?
You can attend any Catholic service any time. I (Hugh) went to a Catholic Church for seven years before becoming Catholic (yeah, I know, it's a guy thing, fear of commitment). The only thing is please don't take communion until you are in communion. That's a big deal for Catholics, because we feel responsible for the Eucharist and we want to make sure it is being received with the right understanding and conditions.
Many Catholic Churches offer a blessing where you can join the Communion line and put your arms across your chest in an "X", so the priest can pray a blessing. I just used communion time for prayer which was fine.
The process of becoming Catholic
The most important thing is prayer. Pray for God to steer you to the right people who can help you make this transition. I pray you meet Holy Spirit-filled, devout, Bible-reading Catholics, of which there are many. After offering a sincere prayer for guidance, my suggestion is to contact a Catholic Church in your area. They can introduce you to the right people to make the transition.
New Catholics - RCIA Classes
Generally, what will happen is that you will join a Rite of Christian Initiation for Adults (RCIA) class that will run from September to Easter one night per week at a parish in your area. During those classes you will cover Catholic teaching, which is basically what you will find on this site. The classes are organized into different subjects. There are no tests or exams. Each week there is a speaker who will come in and give a talk about a particular subject, like Mary, or the Eucharist. There will be anywhere from 2 - 60 participants, but in many cities RCIA classes are about a dozen people. At the Easter Vigil you'd be accepted into the Church.
Perhaps you were brought up Catholic, and some nun swatted you across the head for chewing bubble gum. Yeah, they were pretty strict, but so were all the Protestant churches in those days. Perhaps after leaving the Catholic Church you wandered for a while, drifting into some unhappy romantic situations and stuff, and after experimenting with various types of sin, you came upon Evangelical circles, and had a conversion experience with Jesus there, and now you are re-examining your Catholic roots. This is a very common situation.
We are very sorry that you might not have had a good experience as a Catholic child, and we hope we can make amends. If this is your situation, we suggest you contact a priest in your area. He can help you come back. You would go to confession and restitution, and at that point you'd be back. Although it is not necessary, perhaps you might want to join RCIA classes which are strongly recommended for people coming back to the Catholic Church after a long absence, because it will answer a bunch of those niggly questions you've always had about "why do Catholics do that?"
The RCIA is the normal way that non-Catholic adults become Catholic. However, there are exceptions. Sometimes, if a person can demonstrate that they know Catholic teaching well, and are very serious, things can move quickly. In my case I had been attending Church for 7 years, so the priest gave me a book to read and met with me once a week for about a month. Then I was given a "conditional Baptism" because I could not find my certificate from my Presbyterian infant baptism. So they gave me a Baptism "just in case." Then they did a confirmation. It was an amazing day, and I felt a huge outpouring of graces on that day.
Joining the Catholic Church is like a marriage. Proper preparation is essential, and it's really important on the day of your profession to be very serious about wanting to spend your life with us.
I pray that God will guide you to where He wants you and where you can be of maximum service to his Kingdom. If you decide you want to join us, feel free to drop me an email. Perhaps, I'll see you somewhere as we trudge the road of happy destiny. May God bless you and keep you until then.