The answer to this will vary based on a person's interests, circumstances, knowledge of Islam/Shi'ism, and, possibly, what they are converting from (for instance, were they a Sunni, are they coming from another religion, etc.).
In general, it is good to explore the variety of what Shi'ism and the Shi'i community has to offer - for instance, attending different Shi'i mosques, and listening to lectures (online or in person). One of the advantages of being a convert is that you can usually experience various cultural interpretations of Shi'ism rather than being bound to one. (This is sometimes different if someone marries into the faith, in which case sometimes they are expected to be bound to their spouse's culture.)
While you are motivated, it is good to learn what you can; at the same time, there is a wisdom in not overdoing it, so that you do not get burnout.
It is good to recite the major recommended prayers, such as Du'a Kumayl, once a week, either individually or in a group, and to contemplate them.
If possible, it is good to visit one or more of the shrines of the Imams (like Imam Ali, Imam Husayn, or Imam Reza), or Sayyida Zaynab in Syria or Fatima Masumah in Qom.
It is good to have friendship with genuine mu'mineen (people of sincere faith) and to spend time together in person (not only digitally).
You should participate in the major commemorations with other Shi'is, like the first 10 days of Muharram, Ashura, and Laylat al-Qadr. Of course, it is good to attend other commemorations as well, but those are particularly valuable. If you have to travel to attend commemorations because there are none near you, then you should travel to do so, if possible.
Sometimes, the social dynamics among Shi'is are different than the social dynamics among Sunnis, and, if coming from Sunnism, this can be worth reflecting on. Also, the worldview among Shi'is tends to be different than in Sunnism, in ways which you can discover for yourself, if that is your situation.
It may be worthwhile to spend some time (which may be years) considering what it means for you to be Shi'i. For some people, this involves considering what it means regarding their beliefs about the Prophet and Imams, and how they interact with them spiritually. For some people, this may involve a political or activist stance (whereas still others may not see politics as an essential part of their faith). For some people, this involves a heavy interest in jurisprudence.
If you are new to Islam, it is good to become familiar with the Qur'an. It is good to spend some time every day, even just a little time, reading it and thinking about it. Of course, if you listen to Shi'i lectures, you will also become more familiar with the Qur'an.
It is good to read some narrations (hadith) related in Shi'i books. A good general collection to start with is The Scale of Wisdom: A Compendium of Shi'i Hadith (while keeping in mind that some of this book is related from Sunni sources; a specialist can help you if you have any questions).
If you are new to Shi'i ritual practice, such as how to say salat (daily prayers), you can find some introductory books on this website, as well as videos on YouTube.
If you do not speak a major heritage language used by Shi'is (such as Arabic, Farsi, or Urdu), it is worth learning one - although there is more English-language Shi'i expression, a lot of Shi'i expression is done in heritage languages, and so it will allow you to have a deeper experience. Of course, Arabic is always important for Qur'an and hadith.
Also, keep in mind that every community has honest and dishonest people, sincere and insincere people, selfless and self-serving people, etc. Muslims also vary in how much they adhere to Islamic ideals and teachings. While any person who misbehaves is not a true Shi'i, nonetheless, one finds a variety of people among Shi'is, the same as everywhere else. So, while valuing your true friends, be sure to use wisdom and common sense when dealing with other people (especially in matters relating to finances, visas, or marriage), and if something seems wrong, stay alert to it.
(I am mentioning this because sometimes when people convert to Islam, they have an idealized view of Muslims, and while many Muslims have good ethics, exceptions apply.)
That is a lot - you do not need to do it all at once - but they are things you can look into now or over time.
Best wishes on your journey!
