Sorry for the wait. There was a Sabbath.
First of all, I assure that any person can enter the synagogue: Jewish and non-Jewish people. In a large synagogue, they will not even pay attention to you; excursions with curious non-Jews are not uncommon. But you should especially dress: closed shoes, pants, any headgear.
Secondly, maybe you are already a Jew and you just need to start keeping the commandments. According to Halacha, a Jew is one born of a Jewish mother. That's all. Even if your father/grandfather of father is a Jew, you will not be a Jew. But if you are ready to take on the heaviest burden of fulfilling the 613 commandments and are ready to accept the danger of being a Jew, then such people are provided with giyur.
If you're a Halachic Jew, then come to the nearest synagogue. You will be met very hospitably and will surely tell you the basics of Judaism and how to begin to keep its commandments. If you are a non-Jew, then you have two choices: apply to the nearest synagogue with a request for Giyur and begin to observe all the Jewish commandments, or you join the community of the Sons of Noah. There you will learn the Torah, keep the commandments that Hashem gave to non-Jewish people and will talk with the same non-jewish people as you. In any case, the rabbi to whom you ask for giyur will most likely send you to the Sons of Noah community first.
Conversioning to Judaism - is…
Religion: Start observing all Jewish commandments at 100% without any alleviations from yourself.
People: Become mentally and emotionally part of the Jewish people. To be worthy of giyur, you have to get a serious education and the main thing is to change your life. Habits, budget and schedule for the values of the Jewish people. You may even have to change your social circle, place of residence and job.
How long does giyur take? Personally, I rarely saw that someone who put average efforts for a giyur pass it in less than 5-7 years. If you make great efforts and take a very responsible approach to learning, then you can keep within 2-3 years.
I recommend reading the material below if you want to start preparing for the preparation for the giyur (Russian language):http://ravvinat.ru/ru/departments/giyur/giyur-themes/
Book "Jewish world" by Joseph Telushkin.