I think one of the impressive things about this story is that it remains relatable and relevant today. Of course, there's something of an appeal to authority in this argument because the book is often assigned by schools rather than independently selected by readers, and this might make it appear more important in terms of literary significance than it actually is. However, I think the reason that it is continually chosen as a focal point for high school literature is that its themes stand independently of its setting.
Consider that, with a few minor exceptions in detail, the book is not necessarily locked into California in the 1920s, but that this is a way of facilitating familiarity and sympathy among readers, and for modern educators, of lending insight into what life was like during this period. The underlying themes could have been set at any place or time; people seek to fulfill their dreams and avoid loneliness, our destinies are often out of our control, sympathy can be both virtuous and destructive, and fear often makes us irrational. More detail on the various themes is available in the enotes link below.
In short, one of the things I enjoy about Steinbeck as an author is his ability to ennoble the common person; he can accurately depict the full range of human experience by some of the "lowest" members of society, without making them seem pathetic or reprehensible. This is present in his other works as well, such as Grapes of Wrath.
Some connections to our modern world should be self-evident; the themes are universal and can never be resolved or dispelled by technology or wealth. I would imagine that your teacher might be looking for connections to specific modern events as well, such as drawing a parallel to migrant farm workers from Central America. A connection I find particularly interesting is the theme (this is also in the enotes link) of "the need to take action". Our world is more passive than ever before; it's incredibly easy to do nothing of any real meaning, while convincing ourselves that we are worldly and educated. You might take something like Candy's statement about his dog and compare this to one's own responsibilities in the modern world.
There are a large number of quotes, including page numbers, in the second link below. I would suggest using Candy's quote about his dog, and one of George's quotes about responsibility.