The issue is not the baptism, but what you believe. If you believe in Jesus as the savior, and the Trinity, then you are fully Christian even if you had been born Jewish (though you could later renounce that and return to Judaism).
If you believe in Judaism and do not believe in Christianity (or other religions), you are Jewish (though you would also need a formal conversion to Judaism if you had not been born Jewish).
A Jew who underwent baptism but does not believe in Christianity has not changed from being Jewish just because of that ceremony alone.
Other information
Judaism does not proselytize or seek converts, but it does accept sincere converts.
Conversion is a life-changing and very serious undertaking and a potential convert should think it over carefully. It must not be done on a whim or because of temporary circumstances. One who converts is expected (from then on) to live as a Jew.
The first step would be, without outside help or influence, to get to know about Judaism. The person who felt interested should hang around a Jewish community in order to get a sense of whether Jews and Judaism as it is actually lived appeal to him/her. Also to do some reading. English-language books on Jewish thought and life can be found in Hebrew bookstores.
After that, the interested person would approach a Rabbi and ask about enrolling in a relevant formal course of study. This can last for several years and is an integral part of the process, since he/she will have to actually practice the religion.
Once that is completed, the convert immerses in a mikveh (ritual immersion); and, for uncircumcised males, undergoes circumcision.
Yes, a person who has been baptized as a Catholic can also be baptized as a Christian. The Catholic Church recognizes the validity of baptisms performed in other Christian denominations.
I was baptized as a Catholic when I was a baby.
You are considered a Jew if your mother was a Jew. You are a Catholic if you were baptised in a Catholic Church. So it is entirely possibly to be a Jew who is a Catholic.
Yes, a Catholic child should be baptized in a Catholic church.
Yes, she was baptized as a Catholic while still an infant.
There is no minimum or maximum age to be baptized in the Catholic Church.
Catholics do not 'take' communion, they 'receive' communion. Yes, you must be a baptized Catholic to receive communion in a Catholic Church. Also, you must have undergone instruction and received your First Holy Communion.
Yes, being baptized in the Catholic faith typically means that you are considered a Catholic for life.
No you are not a born catholic but you can be born into a family that is catholic. To become catholic you must be baptized
To take away the sins of the world they were born into. Adults can be baptized also if they have not been previously.
To be considered Catholic by the Church, you must be baptized in the Catholic Church, or else properly baptized in another church and formally received into the Catholic Church by a priest. Being born to Catholic parents isn't enough. You can be the Pope's nephew, but if you haven't been baptized, you are not Catholic.
If you wish to be a Catholic, yes you will need to be baptized, unless you have received a valid baptism elsewhere.