While modern Jews do follow the calendar of whatever country they live in, they may also notice the Jewish calendar - both days and months - which works very differently from the Christian/secular calendar.
The Jewish Day
Unlike the western world (and Christianity), Judaism counts days as starting/ending at sunset. This is due to the book of Genesis - when God creates the world, each day is listed as “And there was evening, and there was morning—the [ordinal] day.” Consequently, a Jewish day starts with evening and ends with late afternoon.
I’m sure you notice the obvious problem: sunset is not only at different times throughout the world, it also changes daily.
The Rabbis allowed for that when codifying how Jews observe the Sabbath. Because these are not set times and you want to include all of a holiday in the day, Shabbat (and all other holidays) starts at the beginning of sundown and ends at the end of sundown, a difference of somewhere between 45 and 90 minutes, depending on your observance, but usually averaging around 70. Basically, we’re going a little over in both directions to make sure we’re definitely observing all of it.
If you’re in a part of the world with no day or no night (above the arctic circle or below the antarctic circle), the rules are confusing and to some degree it’s “choose a method and stick with it.” Judaism was invented in Israel, which…does not have that issue.
The Jewish Calendar
The main thing to know about the Jewish calendar is that it is a lunisolar calendar. The Christian/secular calendar is a solar calendar, with a year determined by the amount of time it takes to move around the sun and the months divided approximately equally into that time. The Islamic calendar is a lunar calendar, with a month determined by the amount of time it takes for the moon to go around the earth, and a year being twelve months, which means the year is only 354-5 days and ends earlier each solar year.
In comparison, a lunisolar calendar, such as the Jewish calendar or the Chinese calendar, determines months by the moon, but a year can be either twelve or thirteen months - with the goal being that the same months will fall during approximately the same seasons. So Rosh Hashanah is always in the fall, Hanukkah is always in the winter, and Passover is always in the spring.
The Jewish calendar has twelve months twelve years out of every nineteen and thirteen months seven years out of every nineteen. Leap years (in which an extra month, not day, is added) happen approximately every third year, though there are two points in the cycle where a leap year will be in the second year after another leap year. On top of that, Jewish holidays can only fall on certain days of the week by rabbinic decree (from what I’ve been told, it’s to avoid going straight from certain holidays into Shabbat and straight from Shabbat into certain holidays), so sometimes the months get moved slightly to make it line up.
The months are:
If there is a leap year, Adar is the month which is duplicated. It becomes Adar Aleph and Adar Bet.
For pronunciation of the months, the best resource I have found is Shalom Sesame (aka the Hebrew version of Sesame Street).
In general, your characters may or may not know what Hebrew year it is (as I write this in August 2018 it is currently 5778, which will change in September at Rosh Hashanah). That’ll depend on their level of Jewish education/investment/interest.
They’ll also most likely have to check exactly what days holidays will fall on, though they’ll know the general time of year. Luckily, nowadays we have websites that will calculate that in an instant - I remember as a kid having a big hardcover book that we used to have to look up dates in, especially historical or future ones.
Basically, the main takeaway here is that no one is ever going to remember every date, and Hebcal is an incredibly useful website for calculating when holidays would be for your characters in any specific year.