Amina Inloes is originally from the US and has a PhD in Islamic Studies from the University of Exeter on Shi'a hadith. She is the program leader for the MA Islamic Studies program at the Islamic College in London and also the Managing Editor of the Journal of Shi'a Islamic Studies.
Shaykh Dr Zaid Alsalami is an Iraqi born scholar, raised in Australia. He obtained a BA from Al-Mustafa University, Qom, and an MA from the Islamic College in London. He also obtained a PhD from ANU, Canberra. He has written and translated several Islamic texts and also prepared educational videos on Islamic rulings and practices.
Mecca, Medina, and Karbala are all considered holy sites in Shi'i narrations and tradition.
It is not really useful to try to rank them. Mostly people ask this for sectarian reasons, i.e. to try to show whether or not Shi'is have the "correct" belief for a Muslim, or to judge whether individual Shi'is have the "correct" belief.
Each of these sites has its own role in the history of Islam and also has its own spiritual character.
Furthermore, individual Shi'is vary in their personal views or personal perceptions of spiritual matters, such as their experience of a place.
However, Mecca is central because it is the place for the required hajj, the direction of prayer, the homeland of the Prophet where he began his mission, and the sacred haram which is visited by the angels and sanctified in the Qur'an. It is also the birthplace of Imam 'Ali and is said to be the place where Adam and Eve first came to earth.
Medina is also important as the site of the first Muslim community, the site of the final resting place of the Prophet, and the residence of some of the Imams.
According to Shi'i law, it is necessary (wajib) to perform pilgrimage to Mecca, and according to Shi'i narrations, one should also visit the Prophet's final resting place in Medina. It is considered highly recommended but not shari'ah-wajib to visit Karbala. So this gives some ranking.
It is also narrated from Imam al-Sadiq that every evening, seventy thousand angels descend, circle the Ka'bah, then go to visit the graves of Imam 'Ali and Imam Husayn, and then return to the heavens. So this also gives some idea of ranking.
Karbala is also emphasized in the Shi'i tradition, both for its spiritual merits, and also because the martyrdom of Imam Husayn is seen as central to the survival of Islam.
There are a handful of narrations in which, on a creational or supra-worldly level, Karbala is treated as higher than Mecca. However, there are also narrations from the Shi'i Imams that say things like "Allah has preferred Mecca over all other cities" and "one prayer in the Prophet's Mosque [in Medina] is equal to a thousand prayers prayed elsewhere."
So, from this, perhaps we can understand that the intent in these narrations is not to rank sacred sites, but just to indicate that certain places have a special sanctity.
Today, some Shi'is individually may say that, for them, visiting Karbala was a more spiritual experience than performing the hajj. This is likely due to the regrettable materialism that has overtaken Mecca (the clock tower, McDonald's, luxury hotels and meals for those who can afford it, etc), the demolition or alteration of sacred sites (such as the jamarat or Jannat al-Baqi'), mismanagement, the crowds, and the hostility towards Shi'is in today's Saudi, including from the minbar and by the religious police.
In any case, Shi'is worldwide are dedicated to performing the hajj, visiting Medina, and also visting Karbala.