Baptism, in the Catholic Church, is sometimes called "the gateway to the Sacraments":
Holy Baptism is the basis of the whole Christian life, the gateway to life in the Spirit (vitae spiritualis ianua), and the door which gives access to the other sacraments.
Catechism of the Catholic Church paragraph 1213
Baptism [is] the gateway to the sacraments and necessary for salvation...
Code of Canon Law 849
You can't, for example, be confirmed without being baptized, and you can't be ordained without being confirmed—and of course you can't be married or ordained if you've already received either sacrament. So it's crucial to be able to see whether someone was baptized or not.
For this reason, records of baptism are very important in church life, and are required by canon law to be kept at the parish and diocesan levels. The Code of Canon Law has a whole chapter dedicated to the record-keeping process. Central to this is canon 877, section 1:
The pastor of the place where the baptism is celebrated must carefully and without any delay record in the baptismal register the names of the baptized, with mention made of the minister, parents, sponsors, witnesses, if any, the place and date of the conferral of the baptism, and the date and place of birth.
The Code doesn't specify exactly how quickly "without any delay" must be, but 24 to 48 hours seems reasonable in most cases.
The parish records are to be communicated to the diocese:
A diocesan bishop is to take care that the acts and documents of the archives of cathedral, collegiate, parochial, and other churches in his territory are also diligently preserved and that inventories or catalogs are made in duplicate, one of which is to be preserved in the archive of the church and the other in the diocesan archive.
(Canon 491, Section 1)
There's no need or legal requirement for these records to be copied to the Vatican. The Church is less centralized than many people think in that regard.
As far as whether someone could look up to see whether you were married, for example: briefly, yes. Before receiving a sacrament, you (or your guardian) has to provide the parish with your baptismal certificate. Because of the diocesan reporting requirements, you can find this even if you don't know what parish you were baptized in.
Once you are married, or have your First Communion, or whatever, that is noted in your local parish records, and a note is also sent to your baptismal parish. They in turn will annotate your baptismal certificate with a record of the sacrament. That way, if you're receiving a new sacrament, you just need your original baptismal certificate. You don't have to go looking for everywhere you might have had a sacrament administered.
There's only sacramental information on the certificate, though. The Church needs to know what sacraments you've been administered. They don't need to know whether you've gone to Catholic school, or whether you're in the Knights of Columbus, or anything like that.
So: Any parish where you have received a sacrament will have a record of that event. So will the diocese of that parish. And your baptismal parish and diocese will have records of all the sacraments you've ever been administered (including First Confession and First Communion). The Vatican won't have any records of any of these. And no one will have records of non-sacramental actions of yours.