According to Judaism there is no such a thing as a half Jew, you are considered %100 a complete Jew if your mother was born Jewish (or converted to Judaism according to halacha before you were born) even if she later strayed away from Judaism and its practices , on the other hand if your mother was not Jewish then you are not Jewish and need to undergo a conversion to join the Jewish people.
According to what you describe it appears that you are totally Jewish.
So welcome back home! it is very special to be part of the Jewish nation the chosen people yet together with this great merit there are also many special obligations which g-d has commanded us (mitzvot).
I strongly recommend you now study more, learn about your special heritage and to learn to mitzvot commandments in order to properly fulfil them.
Addendum
I would like to bring to your attention some of basics of Judaism.
it is of strong importance to know the primary Mitzvot – commandments in order to truly act as a proper Jew.
Important Mitzvot
Kashrut
Kosher as the Jewish people are the chosen nation g-d has given us a very special diet of what to eat in order to retain our sanctity and purity.
Highlights of kosher:
Meat or chicken can only be from kosher animals (cow, sheep, goat, chicken or turkey and not pig, horse).
Additionally, it and needs to be slaughtered in a special manner which can only be done by a ritual slaughter a shochet and that why you cannot purchase meat at any grocery store.
Fish can only be from a fish which has fins and scales (carp, salmon, turnpike, tuna and not lobsters, shrimps or dolphins).
Meat products cannot be eaten together with dairy products (after eating meat one is to wait six hours to eat dairy).
Shabbat
As the Jewish nation is sanctified and chosen g-d gave us the merit to partake in his rest day the seventh day the Shabbat
Highlights of shabbat
We light candles on its onset as a sign that the special day is ushering in
Once the shabbat enters we sanctify the day with the kiddush on a cup of wine and we break bread with two loaves of bread remembering the double portion of manna our forefathers received in the desert in honor of Shabbat
As Shabbat is a rest day, we try to spend all our time involved in holiness and spirituality it is very important to go to synagogue on shabbat and join the special prayers which are recited on this day.
and we are obligated to refrain from work on this day , no telephone or computers we don’t drive cars or even turn on lights or air conditioners (we can leave them on before shabbat or set them with a timer), We do not use any electronic appliances as well.
if we walk outdoors, we are allowed to carry anything in our pocket or hands (unless the area is fenced in or has a eruv).
Family purity
As our body is holy and precious there are special rules of when and how a Jew can and should fulfil his marital relations which otherwise this would be considered a mundane act the Jew has the ability to insert holiness also in the physical, and with in mind the Jewish women is to immerse in a mikveh when she is required to according to Jewish law and only thereafter be with her husband.
This was a short synopsis of three important mitzva’s which surround our lives all the time, again it is important to advance in the observing of the mitzvot to truly live up to our responsibility and thereby allow our soul to shine.
