I am in the final course of my bachelor degree program. I am from the country of Georgia.
I am aiming to complete a PhD in USA without a master's degree in a mathematics field.
So I have one question about it:
Yes, it's normal for PhD students in the US to travel to visit family. You'll have to pay all costs yourself. You'll want to pay careful attention to any restrictions with your visa, and consider complications of international travel that may make it difficult for you to either return home or get back to the US. Certainly international conflicts and sanctions, as well as health-related travel restrictions as have been experienced along with the COVID-19 pandemic are all concerns, but these are not particularly specific to being a graduate student.
PhD students in the US are most often funded by doing some sort of work while they study; for math, in particular, this is often through appointment as a teaching assistant. Being a TA means that travel during the semester is problematic, as you are likely expected to be present for all course sessions and avoidable reasons for absence (like personal travel) are frowned upon.
University-level education in the US is not monolithic - every institution will have their own policies and structure, but a common structure is to have a fall semester from September to early/mid December, followed by a break of a month or so, with courses resuming for Spring semester mid/late January and continuing to early May. Undergraduates may have a week off during Spring (typically March or April), and a few days around US Thanksgiving (4th Thursday of November). Summers can be complicated funding- and responsibility-wise, as you may either more or less have the summer "off" or may be busy teaching an additional semester.
PhD students do not typically have specified "vacation days" the way ordinary employees do, but you'll want to coordinate your vacation time with your PhD advisor at least. This is certainly a question you should ask when interviewing with prospective graduate programs: "I would like to visit my family in my home country during my PhD; what are the opportunities and limits on taking time off to travel?" is a very valid question to ask, and anyone who reacts negatively to such a reasonable request is not someone you want to work with.
I don't know anything about the visa you may be granted to study in the US, so purely a calendar answer:
There are traditionally three vacation periods in the US academic calendar:
The end-of-year holiday break, centered around Christmas, Hanukah and New Year's. This may be a couple of weeks or longer than a month.
Spring break: Usually a week long, sometime in March or April. This can be a great time to escape miserably cold universities.
Summer break: While most PhD students are not "off" in the summer, it is easier to travel because taking and teaching classes is less common. Summer break usually lasts from May or June to August or September. Many conferences take place in the summer, however.
Some universities also give a whole week off for American Thanksgiving or have a week-long fall break, but this is less common. Many Americans travel home for Thanksgiving, which may complicate or even make a trip home easier.
That said, if you are not doing research that requires your presence, or are able to take classes virtually, PhD programs can be very flexible in allowing you to continue your studies and work while not in your city. However, the pay is probably not high enough to allow you to take many trips home (especially since Georgia isn't a huge travel destination for Americans like say, London or Paris).
You specified Georgia as your home country and unfortunately I don't know what kind of student visa you will get but this can be an issue.
From personal experience graduate students from Western Europe had 5 year visas that allowed unlimited entries and exits to the US, so for them there were no visa considerations.
However students from China or Iran only got 1 year visas which include the possibility to stay in the US after the 1 year is over but not the possibility to leave and reenter. This meant that these students had to reapply for a new visa every time they went home. The Chinese students regularly 'took the risk' to go home and most of the time got a new visa without problems, although some had to stay in China longer than they intended before the new visa was available. The Iranian student did not dare to travel home because he was unsure whether he would be granted a new visa.
So carefully check the conditions on your visa before you do travel plans.
Edit from comments: It seems to be official policy that Georgians students should get 5 year visa. If you indeed have such a visa you do not have to worry about this issue.
tl;dr The US makes it significantly more difficult to visit home than most other countries.
In most countries, the visa is the document that grants you the permission to: (1) enter the country and (2) stay within the country. The US is unusual in that these two things are separated. A US visa is only for entering the country. Many students find themselves in the situation that they do have the authorization to stay and study in the US, but their visa is expired. Thus once they leave the country, they must apply for a new visa before they can return.
Citizens of some countries typically get visas with short validity (1 year). In practice, they must apply for a new visa each time they go home. It is not guaranteed that the new visa will be granted. Therefore, the visa application can be stressful, and can take time away from your holidays.
I have seen cases where the visa was initially denied, and the student was unable to return to the US for some time.
I have seen cases where people did not leave the US because of the difficulty and risk in obtaining a new visa.
Yes, there are holidays. Exactly when they are depends on the school.
The biggest problem for international students is the possibility that your visa will be revoked or your home country will not allow you to return to the US. But these are not questions that can be answered here. You need to know what the current diplomatic status is of Georgia with the US and whether that is likely to change in the future.
I have no experience with US programs, so I base my answer here on information in the sample of leave policies in this related question. A standard PhD program provides annual leave that can be used at the discretion of the candidate. You can review the PhD rules at specific universities of interest to you to see how much annual leave they provide. In the linked question there is some discussion of the average amount of annual leave in a PhD program in the US, so it may aid your understanding of common levels of entitlement.
Additionally, in current circumstances it is becoming common in the professions to have "hybrid work" where workers do some of their work remotely. Depending on your field of study and your advisor, it might be possible for you to do some of your candidature remotely, at least for small periods. Once you have an advisor in mind, you can inquire into options for working on your candidature remotely for relatively small periods to augment your leave entitlement and give you some additional time with family. Bear in mind that this option would depend heavily on whether your advisor approves, so it is really a conversation that you would need to have with specific advisors during the application process; you should also be careful to ensure that potential advisors are aware of your overall commitment to the candidature.
