I do think it is important for us to try to understand/know different cultures
The answer is "understand".
The phrases "try to figure something out" and "try to understand something" are idiomatic. They are common expressions, and correct.
On the other hand, "try to know" is not idiomatic in the same way. For example "I am trying to know what to do" doesn't sound right. In many cases, you either "know" or "don't know", but you don't "try to know". Rather, you would try to figure it out, or try to learn, so then you could say "Now I know it."
You could say "Trying to get to know somebody".
If you remove "try", the choices are:
I do think it is important for us to understand/know different cultures
Even here, "know" is not exactly right. It should be "know about".