First let me clarify; there are two identities that are both commonly known as Jewish. One is the genetic identity that is a descendant of the biblical Jacob (who's name was changed to Israel in Genesis 32:28) by one of two wives (Leah and Rachel) or one of two concubines (Bilhah and Zilpah) by which Jacob/Israel fathered twelve sons who became the twelve tribes of Israel. Those tribes are: Reuben, Simeon, Levi, Judah, Dan, Naphtali, Gad, Asher, Issachar, Zebulun, the house of Joseph (Manasseh and Ephraim) and Benjamin.
To clarify the difference, and since Jew means "from Judah" - Judah being only one of twelve tribes - I'll use the term Israelite to refer to the genetic relation. I keep wanting to use Israeli but that would be a member of the modern state of Israel.
The second identity that is commonly known as Jewish is the religious identity. Simply put, it is the religion of the children of Israel. The stories of the founding fathers as well as the laws and customs of the religion are found in the five Books of Moses, which are the first five books of the bible (Genesis/Bereshit, Exodus/Shmot, Leviticus/Vayikra, Numbers/Bamidbar and Deuteronomy/Devarim.)
If you do, and I suggest that you do, look into your ancestral Jewish roots part of that would be reading the Books of Moses. In which you'll find all of the biblically based laws and customs that - I think - you need to know to follow God's will as laid out in the Torah and Old Testament.
Again, to clarify the difference, I'll use the term Judaism to reference the religious identity.
So to recap: when I say "Israelite" I'm referring to that genetic relation to Jacob while if I say "Judaism" I'm referring to the religious relation to Jacob.
The big difference, the key difference between Christianity and Judaism is Jesus. Christians believe that Jesus is the Messiah of Jewish prophecy and that He came, lived, fulfilled all prophecies, died for our sins, rose again triumphant over death, hell and the grave, etc. Followers of Judaism believe that Jesus was a great teacher but that they're still waiting for Messiah to come. There's the whole Sabbath controversy (Saturday Sabbath versus Sunday Sabbath) and keeping kosher but even then there are several forms therein Judaism (Orthodox, Reform, Hasidic, Reconstructionist, Rabbinic, etc.) as well as denominations in Christianity that teaches keeping kosher, Shabbat (worshiping on Saturday) and the festivals as biblically founded. One of these is called Messianic Christianity, Messianic Judaism or commonly the Messianic Movement. And it might be a really great way to honor your heritage without renouncing Christ.
My suggestion is that you look into the Messianic movement because they embrace and celebrate the heritage of Judaism while having the belief of Jesus as Messiah and Savior.
As for your statement of being "not really into organized religion," in spite of that, it sounds like God is calling you. And as one who was raised in a pew - actually quite literally - I can more than empathize with the concept of weariness on the subject of religion. It's important for you to not confuse being forced to sit through five hours of preaching that went over your head with a relationship with God. They're two different things. But, it is also important to keep in mind that we're to "not forsake the assembling of ourselves together." (Hebrews 10:24, 25) And it's very important to find a church that meets your needs. Different churches have different personalities; some are family oriented, some are children oriented (and yes, that's different,) some are youth oriented, some are elder oriented, lots are baby boomer oriented and I suggest that you find one that meets your needs or else it's kind of pointless and pretty frustrating. I also suggest that you seek God's advice on where to go. He does have an opinion. lol
As for your grandmother, please don't get insulted but I'm going to suggest that she might have grown up in a time and place where being Jewish (genetically and/or relgiously) was considered a very bad thing. I know that in my own genealogy that there is Native American blood on my maternal side but we're unaware of who, where and how much because it was considered something to be ashamed of and ignored by those who held the answers to our questions. I've always really wanted to know but the matriarch who knew those answers has since passed away and so now we don't know.
I hope I helped even if only a little.