Answer: determine which branch of Judaism is calling to you – Orthodox, Conservative, Reconstructionist or Reform. Find a synagogue of that branch near you and talk with the rabbi.
Typically, there is an Intro to Judaism class that can last 16 lessons (with a break in the winter or spring, depending on when the class starts). During that time you’ll be expected to attend Shabbat services and holiday observances at the synagogue and bringing more and more of the mitzvot (commandments) into your life, moving closer and closer to living life as a Jew.
After the class you’ll work one-on-one with the rabbi or an elder on any areas you might need more study. When the rabbi considers you are ready, you will be referred to a rabbinical court (who aren’t out to sink anyone). They determine if the candidate understands enough about Judaism and is living life as a Jew (except for saying the blessings on Shabbat).
If the candidate is male and approved, they will have to been circumcised or a ritual drop of blood drawn from the genitals (with appropriate numbing agent applied). Men and women converts then go to a mikvah, a ritual immersion in water with appropriate prayers. When one emerges, they are considered 100% Jewish.
Now, the different branches don’t all recognize conversions by other branches (much like some Christian denominations don’t consider other denominations as REAL Christians).
Orthodox – recognize only Orthodox conversions.
Conservative – recognize Conservative and Orthodox conversions
Reconstructionist – recognize all conversions
Reform – recognize all conversions.
The State of Israel recognizes all conversions as legitimate but only Orthodox Jews can marry in Israel or be buried in a Jewish cemetery. Non-Orthodox go to Cyprus, get married and their marriage is then recognized in Israel.
Orthodox = 2 – 3 years to convert
Conservative = 2 – 2 1/2 years
Reconstructionist – 2 – 2 1/2 years
Reform – 1 – 1 1/2 years
Side note: “Messianic Jews” are NOT Jewish and are NOT accepted in any branch of Judaism. They are Christian sects with Christian beliefs and some are intentionally deceptive in their attempts to convert Jews. “Messianics” are NOT recognized as Jewish by any Jewish branch nor by Israel nor by the US military. They are Christians despite calling their churches synagogues and their ministers “rabbis”
Recommended books: The Complete Idiot's Guide to Understanding Judaism (a real book by a rabbi); Choosing Judaism; Living Judaism; Jewish Literacy; What is Special about Judaism; What is a Jew; The Jewish Home (Amazon.com used books is a great resource - odds are these or most of these will be required in your class and the synagogue can order them for you)