I'm not a teacher, but "get" fits your requirement quite nicely. "I don't understand" = "I don't get it". "Do you get it?" ="Do you understand?" "My wife doesn't get my jokes" = "my wife doesn't understand my jokes".
appreciate, apprehend, catch, catch on (to), dig [slang], get, grasp, make, make out, perceive, see, seize, tumble (to), understand
I'm just a little confused as to what it is you're looking for. Initially you said that you wanted an informal synonym for "understand". Now you say that "get it" is too informal, but that you want something equal to it in formality. Could you clarify what it is you're needing?
I was told that I shouldn't say "I didn't get it" on a memo to my boss and I am not sure about using "I don't comprehend" given that I don't see people using it very often.
English is a powerful language (with more than twice the number of words of my native one) and I just want to improve my vocabulary.
I think what I need is "to realize".