Question: What are the duties of a new convert to Judaism
Answer: Thanks for writing and welcome to the Jewish People. It would appear that you are a very sincere addition to our heritage, and I hope and pray that all you sought in your studies and conversion will be fulfilled for you.
The first resource for a summary of expectations of you should be the rabbi who taught you or officiated your conversion. In my own experience teaching potential converts, a discussion about post-conversion expectations was always part of the educational experience, but perhaps you missed that particular class. I taped all my classes, with the belief that each class was vital, so that anyone who couldn't attend could still learn the material later.
If your teacher or rabbi is distant because you have moved, don't hesitate to use email or a phone call. We as teachers are highly complimented by students continuing to ask us questions and stay in touch; it's never an imposition or "bother."
With regard to the substance of your question, I would say that the first element is continuing education - take every opportunity to take classes at your synagogue and in the community. Just remember that other Rabbis will have other perspectives perhaps and you need to keep your own balance as your grow in your Judaism.
Traditionally, I have always encouraged my students and afterward as full Jews by Choice to affiliate fully with a synagogue and attend worship services/minyanim regularly in the synagogue and to pray at home as well. You may be included in a minyan in your synagogue, and thus you will be needed in a house of mourning for purposes of a week of worship.
There are daily blessings - in addition to before and after food - which should become part of your regular life-style.
Continue to study Hebrew such that reading the Siddur and Mahzor becomes easier and becomes more comprehensible with increasing speed and fluency.
Learn to read from the Torah and the Haftarah, if that is acceptable in your synagogue. It is a skill, at the least, and it also may give you a chance for greater participation in your congregation.
Observance of the dietary laws is extremely important, in and out of the home, while certainly your home is the best place to start.
Giving and participating in charitable enterprises is critical. It is also important to do hands-on Jewish caring for the needs of others. Hands-on can include being part of the synagogue Bikur Holim, that is the group that supports the clergy by visiting and staying in touch by telephone with those who are ill, elderly, alone and shut-ins at home for medical reasons.
When it comes to giving money, be careful as there are a variety of "charities" that are as much self-serving as serving others. UJA-Federation, investing in Israeli bonds, support of Jewish Family and Childrens' Services, Jewish National Fund - trees and land and water development in Israel, are just a few to begin to mention, and don't forget to support your own synagogue.
Lastly, by tradition you should seek out a Jewish husband and build a Jewish home and raise a Jewish family, celebrate the Sabbath and Festivals with home-made hallah (I have a great recipe and there are many!!), and after marriage observe the laws of family purity - again working with your teacher and Rabbi.
Whew! It's a long list. So you may ask how to begin to do so much. My own personal answer is that, especially if all of this is new to you, start where you are comfortable. If you have a significant other, start where both of you are comfortable. Then add to your lives as much as you can, as quickly as you can, without becoming Jewishly burned-out. Finding Jewish friends with whom to celebrate Sabbath day and holidays and lifecycle events is very important and helpful.
Congratulations on asking perhaps the most important question of all - how shall I begin to truly live a Jewish life. For that question, all Jews who may be taking their Judaism for granted should be grateful to you. Good luck!